{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:43:01Z","timestamp":1740134581003,"version":"3.37.3"},"reference-count":8,"publisher":"Wiley","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\/501100001868","name":"National Science Council","doi-asserted-by":"publisher","award":["NSC 103-2115-M-539-001"],"award-info":[{"award-number":["NSC 103-2115-M-539-001"]}],"id":[{"id":"10.13039\/501100001868","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2015]]},"abstract":"<jats:p>Let<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>F<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:msup><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>s<\/mml:mi><mml:mo>-<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:msup><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>s<\/mml:mi><mml:mo>-<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:mo>\u22ef<\/mml:mo><mml:mo>+<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>0<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>be a real-valued polynomial function in which the degree<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:math>of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:math>in<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi>F<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>is greater than or equal to 1. For any polynomial<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mi>y<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>, we assume that<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mi>T<\/mml:mi><mml:mo>:<\/mml:mo><mml:mi mathvariant=\"double-struck\">R<\/mml:mi><mml:mfenced open=\"[\" close=\"]\" separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>\u2192<\/mml:mo><mml:mi mathvariant=\"double-struck\">R<\/mml:mi><mml:mfenced open=\"[\" close=\"]\" separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>is a nonlinear operator with<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mi>T<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>y<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mi>F<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:mi>y<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><\/mml:math>. In this paper, we will find an eigenfunction<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mi>y<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>\u2208<\/mml:mo><mml:mi mathvariant=\"double-struck\">R<\/mml:mi><mml:mfenced open=\"[\" close=\"]\" separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>to satisfy the following equation:<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mi>F<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:mi>y<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mi>a<\/mml:mi><mml:mi>y<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>for some eigenvalue<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mi>a<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi mathvariant=\"double-struck\">R<\/mml:mi><\/mml:math>and we call the problem<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mi>F<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:mi>y<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mi>a<\/mml:mi><mml:mi>y<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>a fixed point like problem. If the number of all eigenfunctions in<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mi>F<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:mi>y<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mi>a<\/mml:mi><mml:mi>y<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>is infinitely many, we prove that (i) any coefficients of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mi>F<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>,<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>,<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>s<\/mml:mi><mml:mo>-<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>,<\/mml:mo><mml:mo>\u2026<\/mml:mo><mml:mo>,<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>0<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>, are all constants in<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mrow><mml:mi mathvariant=\"double-struck\">R<\/mml:mi><\/mml:mrow><\/mml:math>and (ii)<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mi>y<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>is an eigenfunction in<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mi>F<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:mi>y<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mi>a<\/mml:mi><mml:mi>y<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>if and only if<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M17\"><mml:mi>y<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>\u2208<\/mml:mo><mml:mi mathvariant=\"double-struck\">R<\/mml:mi><\/mml:math>.<\/jats:p>","DOI":"10.1155\/2015\/516159","type":"journal-article","created":{"date-parts":[[2015,2,23]],"date-time":"2015-02-23T18:10:04Z","timestamp":1424715004000},"page":"1-6","source":"Crossref","is-referenced-by-count":0,"title":["Infinitely Many Eigenfunctions for Polynomial Problems: Exact Results"],"prefix":"10.1155","volume":"2015","author":[{"given":"Yi-Chou","family":"Chen","sequence":"first","affiliation":[{"name":"Department of General Education, National Army Academy, Taoyuan 320, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1137\/0216040"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.4064\/aa123-2-4"},{"issue":"1","key":"2","first-page":"101","volume":"11","year":"2010","journal-title":"Journal of Nonlinear and Convex 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Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2015\/516159.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2015\/516159.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2015\/516159.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2016,7,27]],"date-time":"2016-07-27T20:11:58Z","timestamp":1469650318000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2015\/516159\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015]]},"references-count":8,"alternative-id":["516159","516159"],"URL":"https:\/\/doi.org\/10.1155\/2015\/516159","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2015]]}}}