{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:13:23Z","timestamp":1758824003399,"version":"3.41.2"},"reference-count":19,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2010,3,22]],"date-time":"2010-03-22T00:00:00Z","timestamp":1269216000000},"content-version":"vor","delay-in-days":80,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"name":"Guangdong College Yumiao"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>This paper studies the existence of multiple solutions of the second\u2010order difference\nboundary value problem <jats:italic>\u0394<\/jats:italic><jats:sup>2<\/jats:sup><jats:italic>u<\/jats:italic>(<jats:italic>n<\/jats:italic> \u2212 1) + <jats:italic>V<\/jats:italic><jats:sup>\u2032<\/jats:sup>(<jats:italic>u<\/jats:italic>(<jats:italic>n<\/jats:italic>)) = 0, <jats:italic>n<\/jats:italic> \u2208 <jats:italic>\u2124<\/jats:italic>(1, <jats:italic>T<\/jats:italic>), <jats:italic>u<\/jats:italic>(0) = 0 = <jats:italic>u<\/jats:italic>(<jats:italic>T<\/jats:italic> + 1). \nBy applying Morse theory, critical groups, and the mountain pass theorem, we prove\nthat  the previous equation\n has at least three nontrivial solutions when the problem is resonant at the\neigenvalue <jats:italic>\u03bb<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>\u2009\u2009(<jats:italic>k<\/jats:italic> \u2265 2) of linear difference problem\n<jats:italic>\u0394<\/jats:italic><jats:sup>2<\/jats:sup><jats:italic>u<\/jats:italic>(<jats:italic>n<\/jats:italic> \u2212 1) + <jats:italic>\u03bb<\/jats:italic><jats:italic>u<\/jats:italic>(<jats:italic>n<\/jats:italic>) = 0, <jats:italic>n<\/jats:italic> \u2208 <jats:italic>\u2124<\/jats:italic>(1, <jats:italic>T<\/jats:italic>),\n<jats:italic>u<\/jats:italic>(0) = 0 = <jats:italic>u<\/jats:italic>(<jats:italic>T<\/jats:italic> + 1)\nnear infinity and the trivial solution of  the first equation is a local minimizer under some assumptions\non <jats:italic>V<\/jats:italic>.<\/jats:p>","DOI":"10.1155\/2010\/907453","type":"journal-article","created":{"date-parts":[[2010,3,22]],"date-time":"2010-03-22T15:30:40Z","timestamp":1269271840000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Existence of Multiple Solutions of a Second\u2010Order Difference Boundary Value Problem"],"prefix":"10.1155","volume":"2010","author":[{"given":"Bo","family":"Zheng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huafeng","family":"Xiao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2010,3,22]]},"reference":[{"doi-asserted-by":"publisher","key":"e_1_2_6_1_2","DOI":"10.1016\/j.na.2003.11.012"},{"doi-asserted-by":"publisher","key":"e_1_2_6_2_2","DOI":"10.1016\/j.amc.2005.12.018"},{"doi-asserted-by":"publisher","key":"e_1_2_6_3_2","DOI":"10.1016\/j.jmaa.2005.07.029"},{"key":"e_1_2_6_4_2","series-title":"Monographs and Textbooks in Pure and Applied Mathematics","doi-asserted-by":"crossref","DOI":"10.1201\/9781420027020","volume-title":"Difference Equations and Inequalities: Theory, Methods & Applications","author":"Agarwal R. 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