{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,1]],"date-time":"2025-11-01T13:56:49Z","timestamp":1762005409039,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,6]],"date-time":"2022-07-06T00:00:00Z","timestamp":1657065600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Liaoning Provincial Department of Education Scientific Research Funding Project","award":["LGKY13217202007","JJKH20221261KJ"],"award-info":[{"award-number":["LGKY13217202007","JJKH20221261KJ"]}]},{"name":"Jilin Provincial Department of Education Science and Technology Research Project","award":["LGKY13217202007","JJKH20221261KJ"],"award-info":[{"award-number":["LGKY13217202007","JJKH20221261KJ"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The solution to a sequential fractional differential equation with affine periodic boundary value conditions is investigated in this paper. The existence theorem of solution is established by means of the Leray\u2013Schauder fixed point theorem and Krasnoselskii fixed point theorem. What is more, the uniqueness theorem of solution is demonstrated via Banach contraction mapping principle. In order to illustrate the main results, two examples are listed.<\/jats:p>","DOI":"10.3390\/sym14071389","type":"journal-article","created":{"date-parts":[[2022,7,6]],"date-time":"2022-07-06T21:15:52Z","timestamp":1657142152000},"page":"1389","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["The Existence and Uniqueness of Solution to Sequential Fractional Differential Equation with Affine Periodic Boundary Value Conditions"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0995-5164","authenticated-orcid":false,"given":"Shanshan","family":"Gao","sequence":"first","affiliation":[{"name":"Department of Basic Teaching, Liaoning Institute of Science and Engineering, Jinzhou 121000, China"},{"name":"College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, 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"}]},{"given":"Cuiying","family":"Li","sequence":"additional","affiliation":[{"name":"College of Mathematical Sciences, Bohai University, Jinzhou 121000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,6]]},"reference":[{"key":"ref_1","first-page":"1208","article-title":"A Study of Generalized Caputo Fractional Differential Equations and Inclusions with Steiljes-type Fractional Integral Boundary Conditions via Fixed-point Theory","volume":"11","author":"Ahmad","year":"2021","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_2","first-page":"1","article-title":"A study of coupled systems of mixed order fractional differential equations and inclusions with coupled integral fractional boundary conditions","volume":"73","author":"Ntouyas","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_3","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. Calc. Appl. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1186\/s13661-020-01418-0","article-title":"Approximate controllability of noninstantaneous impulsive Hilfer fractional integrodifferential equations with fractional Brownian motion","volume":"2020","author":"Ahmed","year":"2020","journal-title":"Bound. Value Probl."},{"key":"ref_5","first-page":"1615","article-title":"Existence of solution to fractional differential equation with fractional integral type boundary conditions","volume":"44","author":"Anwar","year":"2020","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_6","unstructured":"Abdo, M.S., Panchal, S.K., and Bhairat, S.P. (2019). Existence of solution for Hilfer fractional differential equations with boundary value conditions. arXiv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"9915375","DOI":"10.1155\/2021\/9915375","article-title":"New Existence of Solutions for Fractional Integro-Differential Equations with Nonseparated Boundary Conditions","volume":"2021","author":"Ibnelazyz","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1155\/2012\/818703","article-title":"A Note on Fractional Differential Equations with Fractional Separated Boundary Conditions","volume":"2012","author":"Ahmad","year":"2012","journal-title":"Abstr. Appl. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.aml.2015.03.003","article-title":"Explicit iteration and unbounded solutions for fractional integral boundary value problem on an infinite interval","volume":"47","author":"Wang","year":"2015","journal-title":"Appl. Math. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"356","DOI":"10.1016\/j.chaos.2019.04.016","article-title":"Fractional calculus in abstract space and its application in fractional Dirichlet type problems","volume":"123","author":"Peichen","year":"2019","journal-title":"Chaos Solitons Fractals"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1186\/s13662-017-1185-3","article-title":"The positive solutions of infinite-point boundary value problem of fractional differential equations on the infinite interval","volume":"2017","author":"Li","year":"2017","journal-title":"Adv. Differ. Equ."},{"key":"ref_12","first-page":"189","article-title":"Affine-Periodic Solutions for Dissipative Systems","volume":"1","author":"Zhang","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1007\/s10883-018-9425-8","article-title":"Existence of Affine-Periodic Solutions to Newton Affine-Periodic Systems","volume":"25","author":"Xu","year":"2019","journal-title":"J. Dyn. Control Syst."},{"key":"ref_14","first-page":"1624","article-title":"Existence of dissipative-affine-periodic solutions for dissipative-affine-periodic systems","volume":"7","author":"Liu","year":"2017","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1717","DOI":"10.1216\/RMJ-2016-46-5-1717","article-title":"Affine-periodic solutions for nonlinear differential equations","volume":"46","author":"Wang","year":"2016","journal-title":"Rocky Mt. J. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"643","DOI":"10.3934\/dcds.2016.36.643","article-title":"Rotating periodic solutions of second order dissipative dynamical systems","volume":"36","author":"Chang","year":"2016","journal-title":"Discret. Contin. Dyn. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1515\/ans-2015-0113","article-title":"Levinson\u2019s Problem on Affine-Periodic Solutions","volume":"15","author":"Li","year":"2015","journal-title":"Adv. Nonlinear Stud."},{"key":"ref_18","unstructured":"Miller, K.S., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/j.jmaa.2011.05.082","article-title":"Impulsive periodic boundary value problems for fractional differential equation involving Riemann\u2013Liouville sequential fractional derivative","volume":"384","author":"Bai","year":"2011","journal-title":"J. Math. Anal. Appl."},{"key":"ref_20","first-page":"615","article-title":"Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions","volume":"266","author":"Ahmada","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"225","DOI":"10.4067\/S0719-06462021000200225","article-title":"Existence results for a multipoint boundary value problem of nonlinear sequential Hadamard fractional differential equations","volume":"23","author":"Ahmada","year":"2021","journal-title":"Cubo (Temuco)"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1186\/s13662-015-0379-9","article-title":"On Caputo type sequential fractional differential equations with nonlocal integral boundary conditions","volume":"2015","author":"Alsaedi","year":"2015","journal-title":"Adv. Differ. Equ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"29","DOI":"10.53006\/rna.928654","article-title":"Sequential fractional pantograph differential equations with nonlocal boundary conditions: Uniqueness and Ulam-Hyers-Rassias stability","volume":"5","author":"Mohamed","year":"2022","journal-title":"Results Nonlinear Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3046","DOI":"10.1016\/j.camwa.2012.02.036","article-title":"Sequential fractional differential equations with three-point boundary conditions","volume":"64","author":"Ahmada","year":"2012","journal-title":"Comput. Math. Appl."},{"key":"ref_25","first-page":"36","article-title":"On the Existence and Uniqueness Results for Nonlinear Sequential Fractional Differential Equations","volume":"17","author":"Fazli","year":"2018","journal-title":"Appl. Comput. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"111955","DOI":"10.1016\/j.chaos.2022.111955","article-title":"Computational study on the dynamics of fractional order differential equations with applications","volume":"157","author":"Shah","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_27","unstructured":"Podlubny, I. (1998). Fractional Differential Equations, an Introduction to Fractional Derivatives, Elsevier."},{"key":"ref_28","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Frational Differential Equations. North-Holland Mathematics Studies, Elsvier Science B.V."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.aml.2015.11.013","article-title":"Existence of solutions of nonlocal cauchy problem for some fractional abstract differential equation","volume":"55","author":"Deng","year":"2016","journal-title":"Appl. Math. Lett."},{"key":"ref_30","first-page":"123","article-title":"Two remarks about the method of successive approximations","volume":"10","author":"Krasnoselskii","year":"1955","journal-title":"Uspekni Mat. Nauk."},{"key":"ref_31","unstructured":"Dunford, N., and Schwartz, J. (1958). Linear Operators I, Interscience."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/7\/1389\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:43:25Z","timestamp":1760139805000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/7\/1389"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,6]]},"references-count":31,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["sym14071389"],"URL":"https:\/\/doi.org\/10.3390\/sym14071389","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,7,6]]}}}