{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:42:11Z","timestamp":1753886531589,"version":"3.41.2"},"reference-count":12,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,5,9]],"date-time":"2012-05-09T00:00:00Z","timestamp":1336521600000},"content-version":"vor","delay-in-days":129,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"name":"Commission on Higher Education, Thailand"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>By using Krasnoselskii\u2032s fixed point theorem, we study the existence of positive solutions to the three\u2010point summation boundary value problem \u0394<jats:sup>2<\/jats:sup><jats:italic>u<\/jats:italic>(<jats:italic>t<\/jats:italic> \u2212 1) + <jats:italic>a<\/jats:italic>(<jats:italic>t<\/jats:italic>)<jats:italic>f<\/jats:italic>(<jats:italic>u<\/jats:italic>(<jats:italic>t<\/jats:italic>)) = 0, <jats:italic>t<\/jats:italic> \u2208 {1,2, \u2026, <jats:italic>T<\/jats:italic>}, , , where <jats:italic>f<\/jats:italic> is continuous, <jats:italic>T<\/jats:italic> \u2265 3 is a fixed positive integer, <jats:italic>\u03b7<\/jats:italic> \u2208 {1,2, \u2026, <jats:italic>T<\/jats:italic> \u2212 1}, 0 &lt; <jats:italic>\u03b1<\/jats:italic> &lt; (2<jats:italic>T<\/jats:italic> + 2)\/<jats:italic>\u03b7<\/jats:italic>(<jats:italic>\u03b7<\/jats:italic> + 1), 0 &lt; <jats:italic>\u03b2<\/jats:italic> &lt; (2<jats:italic>T<\/jats:italic> + 2 \u2212 <jats:italic>\u03b1<\/jats:italic><jats:italic>\u03b7<\/jats:italic>(<jats:italic>\u03b7<\/jats:italic> + 1))\/<jats:italic>\u03b7<\/jats:italic>(2<jats:italic>T<\/jats:italic> \u2212 <jats:italic>\u03b7<\/jats:italic> + 1), and \u0394<jats:italic>u<\/jats:italic>(<jats:italic>t<\/jats:italic> \u2212 1) = <jats:italic>u<\/jats:italic>(<jats:italic>t<\/jats:italic>) \u2212 <jats:italic>u<\/jats:italic>(<jats:italic>t<\/jats:italic> \u2212 1). We show the existence of at least one positive solution if <jats:italic>f<\/jats:italic> is either superlinear or sublinear.<\/jats:p>","DOI":"10.1155\/2012\/569313","type":"journal-article","created":{"date-parts":[[2012,5,9]],"date-time":"2012-05-09T21:04:01Z","timestamp":1336597441000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Positive Solutions to a Generalized Second\u2010Order Difference Equation with Summation Boundary Value Problem"],"prefix":"10.1155","volume":"2012","author":[{"given":"Thanin","family":"Sitthiwirattham","sequence":"first","affiliation":[]},{"given":"Jessada","family":"Tariboon","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,5,9]]},"reference":[{"key":"e_1_2_6_1_2","first-page":"803","article-title":"Nonlocal boundary-value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects","volume":"23","author":"Ilin V. 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Positive solutions of a second order difference equation with three-point summation boundary value problem Journal of Difference Equations and Applications. In press."},{"volume-title":"Positive Solutions of Operator Equations","year":"1964","author":"Krasnose\u2032ski\u012d M. 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