{"id":"https://openalex.org/W1988295263","doi":"https://doi.org/10.1155/2007/48648","title":"Viscosity Approximation Methods for Nonexpansive Nonself-Mappings in Hilbert Spaces","display_name":"Viscosity Approximation Methods for Nonexpansive Nonself-Mappings in Hilbert Spaces","publication_year":2007,"publication_date":"2007-01-01","ids":{"openalex":"https://openalex.org/W1988295263","doi":"https://doi.org/10.1155/2007/48648","mag":"1988295263"},"language":"en","primary_location":{"id":"doi:10.1155/2007/48648","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2007/48648","pdf_url":"https://downloads.hindawi.com/journals/ijmms/2007/048648.pdf","source":{"id":"https://openalex.org/S151851439","display_name":"International Journal of Mathematics and Mathematical Sciences","issn_l":"0161-1712","issn":["0161-1712","1687-0425"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://downloads.hindawi.com/journals/ijmms/2007/048648.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5071816548","display_name":"Rabian Wangkeeree","orcid":"https://orcid.org/0000-0002-5715-3804"},"institutions":[{"id":"https://openalex.org/I66200439","display_name":"Naresuan University","ror":"https://ror.org/03e2qe334","country_code":"TH","type":"education","lineage":["https://openalex.org/I66200439"]}],"countries":["TH"],"is_corresponding":true,"raw_author_name":"Rabian Wangkeeree","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000","institution_ids":["https://openalex.org/I66200439"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5071816548"],"corresponding_institution_ids":["https://openalex.org/I66200439"],"apc_list":{"value":1025,"currency":"USD","value_usd":1025},"apc_paid":{"value":1025,"currency":"USD","value_usd":1025},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.08447315,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"2007","issue":null,"first_page":"1","last_page":"10"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11777","display_name":"Fixed Point Theorems Analysis","score":0.9876999855041504,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9793000221252441,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.627498984336853},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.39739224314689636},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.3556000590324402},{"id":"https://openalex.org/keywords/database","display_name":"Database","score":0.33735424280166626},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.30884844064712524}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.627498984336853},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.39739224314689636},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3556000590324402},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","level":1,"score":0.33735424280166626},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.30884844064712524}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1155/2007/48648","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2007/48648","pdf_url":"https://downloads.hindawi.com/journals/ijmms/2007/048648.pdf","source":{"id":"https://openalex.org/S151851439","display_name":"International Journal of Mathematics and Mathematical Sciences","issn_l":"0161-1712","issn":["0161-1712","1687-0425"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:hin:jijmms:048648","is_oa":false,"landing_page_url":"http://downloads.hindawi.com/journals/IJMMS/2007/048648.xml","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},{"id":"pmh:oai:doaj.org/article:12ab7378c9194e17a241fe52c1affcfc","is_oa":true,"landing_page_url":"https://doaj.org/article/12ab7378c9194e17a241fe52c1affcfc","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences, Vol 2007 (2007)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/2007/48648","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2007/48648","pdf_url":"https://downloads.hindawi.com/journals/ijmms/2007/048648.pdf","source":{"id":"https://openalex.org/S151851439","display_name":"International Journal of Mathematics and Mathematical Sciences","issn_l":"0161-1712","issn":["0161-1712","1687-0425"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320323020","display_name":"Naresuan University","ror":"https://ror.org/03e2qe334"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1988295263.pdf","grobid_xml":"https://content.openalex.org/works/W1988295263.grobid-xml"},"referenced_works_count":5,"referenced_works":["https://openalex.org/W582182238","https://openalex.org/W1973762341","https://openalex.org/W2033943991","https://openalex.org/W2050197568","https://openalex.org/W2059379442"],"related_works":["https://openalex.org/W2748952813","https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W4245490552","https://openalex.org/W4225152035","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W1587224694","https://openalex.org/W2911598644"],"abstract_inverted_index":{"Viscosity":[0],"approximation":[1],"methods":[2],"for":[3],"nonexpansive":[4,34],"nonself-mappings":[5],"are":[6],"studied.":[7],"Let<mml:math":[8],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[9,19,21,26,28,31,37,39,44,46,48,51,60,68,70,73,77,80,88,95,98,104,120,122],"id=\"E1\"><mml:mi>C</mml:mi></mml:math>be":[10],"a":[11,33,42,82,100],"nonempty":[12],"closed":[13],"convex":[14],"subset":[15],"of":[16,57,92],"Hilbert":[17],"space<mml:math":[18],"id=\"E2\"><mml:mi>H</mml:mi></mml:math>,<mml:math":[20],"id=\"E3\"><mml:mi>P</mml:mi></mml:math>a":[22],"metric":[23],"projection":[24],"of<mml:math":[25,103],"id=\"E4\"><mml:mi>H</mml:mi></mml:math>onto<mml:math":[27],"id=\"E5\"><mml:mi>C</mml:mi></mml:math>and":[29],"let<mml:math":[30,50],"id=\"E6\"><mml:mi>T</mml:mi></mml:math>be":[32],"nonself-mapping":[35],"from<mml:math":[36],"id=\"E7\"><mml:mi>C</mml:mi></mml:math>into<mml:math":[38],"id=\"E8\"><mml:mi>H</mml:mi></mml:math>.":[40],"For":[41],"contraction<mml:math":[43,59],"id=\"E9\"><mml:mi>f</mml:mi></mml:math>on<mml:math":[45],"id=\"E10\"><mml:mi>C</mml:mi></mml:math>and<mml:math":[47],"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>,":[49],"id=\"E12\"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math>be":[52],"the":[53,58,65,93,117],"unique":[54],"fixed":[55,101],"point":[56,102],"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":[61],"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>.":[62],"Consider":[63],"also":[64],"iterative":[66],"processes<mml:math":[67],"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":[69],"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":[71],"by<mml:math":[72],"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":[74],"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>,":[75],"and<mml:math":[76,97],"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":[78],"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":[79],"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":[81],"real":[83,118],"sequence":[84],"in":[85],"an":[86],"interval<mml:math":[87],"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>.":[89],"Strong":[90],"convergence":[91],"sequences<mml:math":[94,119],"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>,":[96],"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":[99],"id=\"E22\"><mml:mi>T</mml:mi></mml:math>which":[105],"solves":[106],"some":[107],"variational":[108],"inequalities":[109],"is":[110],"obtained":[111],"under":[112],"certain":[113],"appropriate":[114],"conditions":[115],"on":[116],"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":[121],"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>.":[123]},"counts_by_year":[],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
