{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:42:00Z","timestamp":1740134520425,"version":"3.37.3"},"reference-count":7,"publisher":"Wiley","license":[{"start":{"date-parts":[[2013,1,1]],"date-time":"2013-01-01T00:00:00Z","timestamp":1356998400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001691","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":[[2013]]},"abstract":"<jats:p>We give an Ulam type stability result for the following functional equation:<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>f<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mrow><mml:mi>\u03b1<\/mml:mi><mml:mi>x<\/mml:mi><mml:mo>\u2212<\/mml:mo><mml:mi>\u03b1<\/mml:mi><mml:msup><mml:mi>x<\/mml:mi><mml:mo>\u2032<\/mml:mo><\/mml:msup><mml:mo>+<\/mml:mo><mml:msub><mml:mi>x<\/mml:mi><mml:mn>0<\/mml:mn><\/mml:msub><\/mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mo>=<\/mml:mo><mml:mi>\u03b2<\/mml:mi><mml:mi>f<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mo>\u2212<\/mml:mo><mml:mi>\u03b2<\/mml:mi><mml:mi>f<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msup><mml:mi>x<\/mml:mi><mml:mo>\u2032<\/mml:mo><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mo>+<\/mml:mo><mml:msub><mml:mi>y<\/mml:mi><mml:mn>0<\/mml:mn><\/mml:msub><mml:mrow><mml:mtext>\u2009\u2009<\/mml:mtext><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mrow><mml:mtext>for all<\/mml:mtext><mml:mtext>\u2009\u2009<\/mml:mtext><mml:mo>x<\/mml:mo><mml:mo>,<\/mml:mo><mml:msup><mml:mi>x<\/mml:mi><mml:mo>\u2032<\/mml:mo><\/mml:msup><mml:mo>\u2208<\/mml:mo><mml:mi>X<\/mml:mi><\/mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>under a suitable condition. We also give a concrete stability result for the case taking up<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mi>\u03b4<\/mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">\u2225<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">\u2225<\/mml:mo><\/mml:mrow><mml:mi>p<\/mml:mi><\/mml:msup><mml:msup><mml:mrow><mml:mo stretchy=\"false\">\u2225<\/mml:mo><mml:msup><mml:mi>x<\/mml:mi><mml:mo>\u2032<\/mml:mo><\/mml:msup><mml:mo stretchy=\"false\">\u2225<\/mml:mo><\/mml:mrow><mml:mi>q<\/mml:mi><\/mml:msup><\/mml:math>as a control function.<\/jats:p>","DOI":"10.1155\/2013\/109754","type":"journal-article","created":{"date-parts":[[2013,3,20]],"date-time":"2013-03-20T17:00:46Z","timestamp":1363798846000},"page":"1-7","source":"Crossref","is-referenced-by-count":0,"title":["The Ulam Type Stability of a Generalized Additive Mapping and Concrete Examples"],"prefix":"10.1155","volume":"2013","author":[{"given":"Hiroyoshi","family":"Oda","sequence":"first","affiliation":[{"name":"Department of Information Sciences, Toho University, Funabashi, Chiba 274-8501, Japan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5405-8900","authenticated-orcid":true,"given":"Makoto","family":"Tsukada","sequence":"additional","affiliation":[{"name":"Department of Information Sciences, Toho University, Funabashi, Chiba 274-8501, Japan"}]},{"given":"Takeshi","family":"Miura","sequence":"additional","affiliation":[{"name":"Yamagata University, Yonezawa, Yamagata 273-0866, Japan"}]},{"given":"Yuji","family":"Kobayashi","sequence":"additional","affiliation":[{"name":"Toho University, Funabashi, Chiba 274-8501, Japan"}]},{"given":"Sin-Ei","family":"Takahasi","sequence":"additional","affiliation":[{"name":"Toho University, Yamagata University, Funabashi, Chiba 273-0866, Japan"}]}],"member":"311","reference":[{"volume":"8","year":"1960","series-title":"Interscience Tracts in Pure and Applied Mathematics,","key":"8"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.27.4.222"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.2969\/jmsj\/00210064"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.2307\/2042795"},{"volume":"19","year":"1966","series-title":"Mathematics in Science and Engineering","key":"1"},{"issue":"3","key":"5","first-page":"423","volume":"11","year":"2010","journal-title":"Journal of Nonlinear and Convex Analysis"},{"issue":"1, article 19","key":"7","volume":"9","year":"2012","journal-title":"The Australian Journal of Mathematical Analysis and Applications"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2013\/109754.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2013\/109754.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2013\/109754.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2016,7,30]],"date-time":"2016-07-30T05:59:49Z","timestamp":1469858389000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/ijmms\/2013\/109754\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013]]},"references-count":7,"alternative-id":["109754","109754"],"URL":"https:\/\/doi.org\/10.1155\/2013\/109754","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2013]]}}}