{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:37:39Z","timestamp":1753886259002,"version":"3.41.2"},"reference-count":8,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2011,12,6]],"date-time":"2011-12-06T00:00:00Z","timestamp":1323129600000},"content-version":"vor","delay-in-days":339,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2011,1]]},"abstract":"<jats:p>We prove the generalized Hyers\u2010Ulam stability of the 2nd\u2010order linear differential\nequation of the form <jats:italic>y<\/jats:italic><jats:sup><jats:italic>\u2032\u2032<\/jats:italic><\/jats:sup> + <jats:italic>p<\/jats:italic>(<jats:italic>x<\/jats:italic>)<jats:italic>y<\/jats:italic><jats:sup><jats:italic>\u2032<\/jats:italic><\/jats:sup> + <jats:italic>q<\/jats:italic>(<jats:italic>x<\/jats:italic>)<jats:italic>y<\/jats:italic> = <jats:italic>f<\/jats:italic>(<jats:italic>x<\/jats:italic>), with condition that there exists a nonzero <jats:italic>y<\/jats:italic><jats:sub>1<\/jats:sub> : <jats:italic>I<\/jats:italic> \u2192 <jats:italic>X<\/jats:italic> in <jats:italic>C<\/jats:italic><jats:sup>2<\/jats:sup>(<jats:italic>I<\/jats:italic>) such that  and <jats:italic>I<\/jats:italic> is an open\ninterval. As a consequence of our main theorem, we prove the generalized Hyers\u2010Ulam\nstability of several important well\u2010known differential equations.<\/jats:p>","DOI":"10.1155\/2011\/813137","type":"journal-article","created":{"date-parts":[[2011,12,6]],"date-time":"2011-12-06T21:10:42Z","timestamp":1323205842000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Generalized Hyers\u2010Ulam Stability of the Second\u2010Order Linear Differential Equations"],"prefix":"10.1155","volume":"2011","author":[{"given":"A.","family":"Javadian","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"E.","family":"Sorouri","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"G. H.","family":"Kim","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M. Eshaghi","family":"Gordji","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2011,12,6]]},"reference":[{"volume-title":"Problems in Modern Mathematics","year":"1940","author":"Ulam S. M.","key":"e_1_2_4_1_2"},{"key":"e_1_2_4_2_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.27.4.222"},{"key":"e_1_2_4_3_2","doi-asserted-by":"publisher","DOI":"10.2307\/2042795"},{"key":"e_1_2_4_4_2","doi-asserted-by":"publisher","DOI":"10.1155\/S102558349800023X"},{"key":"e_1_2_4_5_2","doi-asserted-by":"publisher","DOI":"10.4134\/BKMS.2002.39.2.309"},{"key":"e_1_2_4_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0022\u2010247X(03)00458\u2010X"},{"key":"e_1_2_4_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2003.11.004"},{"key":"e_1_2_4_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2005.11.004"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2011\/813137.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2011\/813137.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2011\/813137","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,10]],"date-time":"2024-06-10T15:03:23Z","timestamp":1718031803000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2011\/813137"}},"subtitle":[],"editor":[{"given":"Kuppalapalle","family":"Vajravelu","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2011,1]]},"references-count":8,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2011,1]]}},"alternative-id":["10.1155\/2011\/813137"],"URL":"https:\/\/doi.org\/10.1155\/2011\/813137","archive":["Portico"],"relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2011,1]]},"assertion":[{"value":"2011-09-26","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2011-10-23","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2011-12-06","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"813137"}}