{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,9]],"date-time":"2025-12-09T15:47:14Z","timestamp":1765295234904,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,9]],"date-time":"2021-07-09T00:00:00Z","timestamp":1625788800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100009555","name":"Henan Normal University","doi-asserted-by":"publisher","award":["71601072, 20B110006"],"award-info":[{"award-number":["71601072, 20B110006"]}],"id":[{"id":"10.13039\/100009555","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We investigate the existence of solutions for a system of m-singular sum fractional q-differential equations in this work under some integral boundary conditions in the sense of Caputo fractional q-derivatives. By means of a fixed point Arzel\u00e1\u2013Ascoli theorem, the existence of positive solutions is obtained. By providing examples involving graphs, tables, and algorithms, our fundamental result about the endpoint is illustrated with some given computational results. In general, symmetry and q-difference equations have a common correlation between each other. In Lie algebra, q-deformations can be constructed with the help of the symmetry concept.<\/jats:p>","DOI":"10.3390\/sym13071235","type":"journal-article","created":{"date-parts":[[2021,7,9]],"date-time":"2021-07-09T10:50:38Z","timestamp":1625827838000},"page":"1235","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Existence of Solutions for a Singular Fractional q-Differential Equations under Riemann\u2013Liouville Integral Boundary Condition"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5450-3127","authenticated-orcid":false,"given":"Mohammad Esmael","family":"Samei","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Basic Science, Bu-Ali Sina University, Hamedan 65178, Iran"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rezvan","family":"Ghaffari","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Basic Science, Bu-Ali Sina University, Hamedan 65178, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5280-2975","authenticated-orcid":false,"given":"Shao-Wen","family":"Yao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2260-0341","authenticated-orcid":false,"given":"Mohammed K. A.","family":"Kaabar","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Washington State University, Pullman, WA 99163, USA"},{"name":"Institute of Mathematical Sciences, University of Malaya, Kuala Lumpur 50603, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Francisco","family":"Mart\u00ednez","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Statistics, Technological University of Cartagena, 30203 Cartagena, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4996-8373","authenticated-orcid":false,"given":"Mustafa","family":"Inc","sequence":"additional","affiliation":[{"name":"Department of Computer Engineering, Biruni University, Istanbul 34025, Turkey"},{"name":"Department of Mathematics, Science Faculty, Firat University, Elazig 23119, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1002\/zamm.19950750820","article-title":"Definition of physical consistent damping laws with fractional derivatives","volume":"75","author":"Beyer","year":"1995","journal-title":"Z. Angew. Math. Mech."},{"key":"ref_2","first-page":"86","article-title":"Some applications of nonlinear fractional differential equations and their approximations","volume":"15","author":"He","year":"1999","journal-title":"Sci. Technol. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/S0045-7825(98)00108-X","article-title":"Approximate analytic solution for seepage flow with fractional derivatives in porous media","volume":"167","author":"He","year":"1998","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1111\/j.1365-246X.1967.tb02303.x","article-title":"Linear models of dissipation whose Q is almost frequency independent-II","volume":"13","author":"Caputo","year":"1967","journal-title":"Geophys. J. Int."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1016\/j.chaos.2005.11.062","article-title":"On chaos synchronization of fractional differential equations","volume":"32","author":"Yan","year":"2007","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1016\/j.bspc.2007.07.013","article-title":"Improvement of the muscle fractional multimodel for low-rate stimulation","volume":"2","author":"Sommacal","year":"2007","journal-title":"Biomed. Signal Process. Control"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"490","DOI":"10.1186\/s13662-020-02952-y","article-title":"SEIR epidemic model for COVID-19 transmission by Caputo derivative of fractional order","volume":"2020","author":"Rezapour","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1007\/s11071-004-3770-8","article-title":"Fractional order control of a hexapod robot","volume":"38","author":"Silva","year":"2004","journal-title":"Nonlinear Dyn."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2421","DOI":"10.1016\/S0165-1684(03)00194-4","article-title":"Fractional differentiation for edge detection","volume":"83","author":"Mathieu","year":"2003","journal-title":"Signal Process"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1006\/jmaa.2000.7194","article-title":"Analysis of Fractional Differential Equations","volume":"265","author":"Diethelm","year":"2002","journal-title":"J. Math. Anal. Appl."},{"key":"ref_11","first-page":"35","article-title":"On q-variant of Dai\u2013Yuan conjugate gradient algorithm for unconstrained optimization problems","volume":"2021","author":"Mishra","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1186\/s13662-019-2414-8","article-title":"Existence and uniqueness of solutions for multi-term fractional q\u2013integro-differential equations via quantum calculus","volume":"2019","author":"Ntouyas","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2677","DOI":"10.1016\/j.na.2007.08.042","article-title":"Basic theory of fractional differential equations","volume":"69","author":"Lakshmikantham","year":"2008","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1186\/s13662-019-2480-y","article-title":"Existence of solutions for a system of singular sum fractional q\u2013differential equations via quantum calculus","volume":"2020","author":"Samei","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1186\/s13662-021-03228-9","article-title":"Investigation of the p-Laplacian nonperiodic nonlinear boundary value problem via generalized Caputo fractional derivatives","volume":"2021","author":"Matar","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"305","DOI":"10.2307\/2370183","article-title":"q\u2013difference equations","volume":"32","author":"Jackson","year":"1910","journal-title":"Am. J. Math."},{"key":"ref_17","first-page":"283","article-title":"The general theory of a class of linear partial q\u2013difference equations","volume":"26","author":"Adams","year":"1924","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_18","first-page":"861","article-title":"Note on the integro-q\u2013difference equations","volume":"31","author":"Adams","year":"1929","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2890","DOI":"10.1016\/j.jfranklin.2014.01.020","article-title":"Existence of solutions for nonlinear fractional q\u2013difference integral equations with two fractional orders and nonlocal four-point boundary conditions","volume":"351","author":"Ahmad","year":"2014","journal-title":"J. Franklin Inst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1186\/1687-1847-2012-140","article-title":"Existence results for nonlocal boundary value problems of nonlinear fractional q\u2013difference equations","volume":"2012","author":"Ahmad","year":"2012","journal-title":"Adv. Differ. Equ."},{"key":"ref_21","first-page":"201","article-title":"Approximate solutions for a fractional q\u2013integro-difference equation","volume":"13","author":"Balkani","year":"2019","journal-title":"J. Math. Ext."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.14232\/ejqtde.2010.1.70","article-title":"Nontrivials solutions for fractional q\u2013difference boundary value problems","volume":"70","author":"Ferreira","year":"2010","journal-title":"Electron. J. Qual. Theory Differ. Equ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1186\/s13660-019-2224-2","article-title":"Existence of solutions for equations and inclusions of multi-term fractional q\u2013integro-differential with non-separated and initial boundary conditions","volume":"2019","author":"Samei","year":"2019","journal-title":"J. Inequalities Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1186\/1687-1847-2013-48","article-title":"Existence results for fractional q\u2013difference equations with nonlocal q\u2013integral boundary conditions","volume":"2013","author":"Zhao","year":"2013","journal-title":"Adv. Differ. Equ."},{"key":"ref_25","first-page":"205","article-title":"Two fractional derivative inclusion problems via integral boundary condition","volume":"257","author":"Agarwal","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1186\/s13662-017-1404-y","article-title":"Approximate solutions of a sum-type fractional integro-differential equation by using Chebyshev and Legendre polynomials","volume":"2017","author":"Rezapour","year":"2017","journal-title":"Adv. Differ. Equ."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1186\/s13662-019-2489-2","article-title":"New approach to solutions of a class of singular fractional q\u2013differential problem via quantum calculus","volume":"2020","author":"Liang","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1186\/s13661-018-1008-9","article-title":"On high order fractional integro-differential equations including the Caputo-Fabrizio derivative","volume":"2018","author":"Aydogan","year":"2018","journal-title":"Bound. Value Probl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1155\/2015\/785738","article-title":"On a time-fractional integro-differential equation via three-point boundary value conditions","volume":"2015","author":"Baleanu","year":"2015","journal-title":"Math. Probl. Eng."},{"key":"ref_30","unstructured":"He, C.Y. (1992). Almost Periodic Differential Equations, Higher Education Press. (In Chinese)."},{"key":"ref_31","first-page":"267","article-title":"Systems d\u2019equations differential d\u2019oscillations nonlinearies","volume":"4","year":"1959","journal-title":"Rev. Roumaine Math. Pure Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1186\/s13662-019-2090-8","article-title":"Existence results for a fractional hybrid differential inclusion with Caputo-Hadamard type fractional derivative","volume":"2019","author":"Samei","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Annaby, M., and Mansour, Z. (2012). q\u2013Fractional Calculus and Equations, Springer.","DOI":"10.1007\/978-3-642-30898-7"},{"key":"ref_34","unstructured":"Samko, S., Kilbas, A., and Marichev, O. (1993). Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers."},{"key":"ref_35","unstructured":"Krasnoselskii, M.A. (1964). Positive Solution of Operator Equation, Noordhoff."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Garrappa, R. (2018). Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial. Mathematics, 6.","DOI":"10.3390\/math6020016"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1235\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:28:17Z","timestamp":1760164097000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1235"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,9]]},"references-count":36,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,7]]}},"alternative-id":["sym13071235"],"URL":"https:\/\/doi.org\/10.3390\/sym13071235","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,7,9]]}}}