{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:54:04Z","timestamp":1760147644902,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,16]],"date-time":"2023-02-16T00:00:00Z","timestamp":1676505600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Jilin Provincial Department of Education Science and Technology, Research","award":["JJKH20221261KJ"],"award-info":[{"award-number":["JJKH20221261KJ"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper is devoted to investigating the existence of solutions for the fractional differential equation and fractional differential inclusion of order \u03b1\u2208(2,3] with affine periodic boundary value conditions. Applying the Leray\u2013Schauder fixed point theorem, the existence of the solutions for the fractional differential equation is established. Furthermore, for the fractional differential inclusion, we consider two cases: (i) the set-valued function has convex value and (ii) the set-valued function has nonconvex value. The main tools of our research are the Leray\u2013Schauder alternative theorem, Covita and Nadler\u2019s fixed point theorem and some set-valued analysis theories.<\/jats:p>","DOI":"10.3390\/sym15020526","type":"journal-article","created":{"date-parts":[[2023,2,16]],"date-time":"2023-02-16T02:28:27Z","timestamp":1676514507000},"page":"526","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Existence Theorems of Fractional Differential Equation and Fractional Differential Inclusion with Affine Periodic Boundary Value Conditions"],"prefix":"10.3390","volume":"15","author":[{"given":"Yan","family":"Wang","sequence":"first","affiliation":[{"name":"Department of Mathematics, Changchun University of Finance and Economics, Changchun 130122, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rui","family":"Wu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Changchun University of Finance and Economics, Changchun 130122, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0995-5164","authenticated-orcid":false,"given":"Shanshan","family":"Gao","sequence":"additional","affiliation":[{"name":"Basic Teaching Department, Liaoning Institute of Science and Engineering, Jinzhou 121000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1007\/s13540-021-00007-x","article-title":"Upper and lower estimates for the separation of solutions to fractional differential equations","volume":"25","author":"Diethelm","year":"2022","journal-title":"Fract. 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