{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T14:39:14Z","timestamp":1648564754873},"reference-count":24,"publisher":"Hindawi Limited","license":[{"start":{"date-parts":[[2015,1,1]],"date-time":"2015-01-01T00:00:00Z","timestamp":1420070400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"crossref","award":["3122014K010"],"award-info":[{"award-number":["3122014K010"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2015]]},"abstract":"<jats:p>Consider the variational inequality<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mtext>V<\/mml:mtext><mml:mtext>I<\/mml:mtext><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>C<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>F<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>of finding a point<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">*<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>\u2208<\/mml:mo><mml:mi>C<\/mml:mi><\/mml:math>satisfying the property<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mo stretchy=\"false\">\u2329<\/mml:mo><mml:mi>F<\/mml:mi><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">*<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>,<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>-<\/mml:mo><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">*<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">\u232a<\/mml:mo><mml:mo>\u2265<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:math>for all<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi>x<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi>C<\/mml:mi><\/mml:math>, where<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><\/mml:math>is a level set of a convex function defined on a real Hilbert space<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mi>H<\/mml:mi><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mi>F<\/mml:mi><mml:mo>:<\/mml:mo><mml:mi>H<\/mml:mi><mml:mo>\u2192<\/mml:mo><mml:mi>H<\/mml:mi><\/mml:math>is a boundedly Lipschitzian (i.e., Lipschitzian on bounded subsets of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:mi>H<\/mml:mi><\/mml:mrow><\/mml:math>) and strongly monotone operator. He and Xu proved that this variational inequality has a unique solution and devised iterative algorithms to approximate this solution (see He and Xu, 2009). In this paper, relaxed and self-adaptive iterative algorithms are proposed for computing this unique solution. Since our algorithms avoid calculating the projection<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mrow><mml:msub><mml:mrow><mml:mi>P<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>(calculating<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mrow><mml:msub><mml:mrow><mml:mi>P<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>by computing a sequence of projections onto half-spaces containing the original domain<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><\/mml:math>) directly and select the stepsizes through a self-adaptive way (having no need to know any information of bounded Lipschitz constants of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mrow><mml:mi>F<\/mml:mi><\/mml:mrow><\/mml:math>(i.e., Lipschitz constants on some bounded subsets of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mrow><mml:mi>H<\/mml:mi><\/mml:mrow><\/mml:math>)), the implementations of our algorithms are very easy. The algorithms in this paper improve and extend the corresponding results of He and Xu.<\/jats:p>","DOI":"10.1155\/2015\/175254","type":"journal-article","created":{"date-parts":[[2015,3,1]],"date-time":"2015-03-01T21:01:20Z","timestamp":1425243680000},"page":"1-7","source":"Crossref","is-referenced-by-count":0,"title":["Iterative Algorithms for Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators"],"prefix":"10.1155","volume":"2015","author":[{"given":"Caiping","family":"Yang","sequence":"first","affiliation":[{"name":"College of Science, Civil Aviation University of China, Tianjin 300300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Songnian","family":"He","sequence":"additional","affiliation":[{"name":"College of Science, Civil Aviation University of China, Tianjin 300300, China"},{"name":"Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"98","reference":[{"key":"1","first-page":"4413","volume":"258","year":"1964","journal-title":"Comptes Rendus de l'Acad\u00e9mie des Sciences"},{"key":"2","year":"1984"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2004.12.023"},{"key":"4","year":"1973"},{"key":"5","year":"1980"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1007\/BF01585696"},{"key":"7","year":"1984"},{"key":"8","year":"2001"},{"key":"9","year":"1981"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1007\/BF01582255"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1023\/A:1013096613105"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2004.04.068"},{"issue":"2","key":"14","first-page":"245","volume":"10","year":"2009","journal-title":"Fixed Point Theory"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719451"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160200302"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1023\/B:JOTA.0000005048.79379.b6"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2004.04.059"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1016\/S0191-2615(96)00030-6"},{"key":"20","first-page":"473","volume-title":"The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed point sets of nonexpansive mappings","year":"2001"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1007\/BF01589441"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.1155\/2013\/942315"},{"key":"23","year":"1990"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/20\/4\/014"},{"key":"25","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/28\/8\/085004"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2015\/175254.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2015\/175254.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2015\/175254.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,15]],"date-time":"2020-05-15T00:27:02Z","timestamp":1589502422000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2015\/175254\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015]]},"references-count":24,"alternative-id":["175254","175254"],"URL":"https:\/\/doi.org\/10.1155\/2015\/175254","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"value":"1110-757X","type":"print"},{"value":"1687-0042","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015]]}}}