{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:53:48Z","timestamp":1760144028834,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,2,29]],"date-time":"2024-02-29T00:00:00Z","timestamp":1709164800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Guangdong Basic and Applied Basic Research Foundation","award":["2022A1515110907"],"award-info":[{"award-number":["2022A1515110907"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The purpose of this paper is to investigate the existence and multiplicity of nontrivial solutions with the \u03c6p-Laplacian for the discrete 2n-th order periodic boundary value issue. To support these conclusions, we have employed variational techniques and contemporary critical point theory. A few new findings are expanded upon and enhanced. We give an example to show how our key findings can be applied.<\/jats:p>","DOI":"10.3390\/axioms13030163","type":"journal-article","created":{"date-parts":[[2024,2,29]],"date-time":"2024-02-29T05:59:19Z","timestamp":1709186359000},"page":"163","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Multiplicity of Solutions for Discrete 2n-TH Order Periodic Boundary Value Problem with \u03c6p-Laplacian"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5858-063X","authenticated-orcid":false,"given":"Jiabin","family":"Zuo","sequence":"first","affiliation":[{"name":"School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Omar","family":"Hammouti","sequence":"additional","affiliation":[{"name":"Laboratory LAMA, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, B.P. 1796 F\u00e8s-Atlas, Fez 30000, Morocco"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Said","family":"Taarabti","sequence":"additional","affiliation":[{"name":"LISTI, National School of Applied Sciences of Agadir, Ibn Zohr University, Agadir 80000, Morocco"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,29]]},"reference":[{"key":"ref_1","unstructured":"Kelly, W.G., and Peterson, A.C. 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Equ."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"232","DOI":"10.2298\/AADM220824008H","article-title":"Anisotropic discrete boundary value problems","volume":"17","author":"Hammouti","year":"2023","journal-title":"Appl. Anal. Discret. Math."},{"key":"ref_13","first-page":"5572645","article-title":"Existence of two positive solutions for two kinds of fractional p-Laplacian equations","volume":"2021","author":"Wu","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P. (2000). Difference Equations and Inequalities, Theory, Methods, and Applications, Marcel Dekker. [2nd ed.].","DOI":"10.1201\/9781420027020"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Rabinowitz, P.H. (1986). Minimax Methods in Critical Point Theory with Applications to Differential Equations, American Mathematical Society.","DOI":"10.1090\/cbms\/065"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/3\/163\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:07:09Z","timestamp":1760105229000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/3\/163"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,29]]},"references-count":15,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["axioms13030163"],"URL":"https:\/\/doi.org\/10.3390\/axioms13030163","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,2,29]]}}}