{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,8]],"date-time":"2026-07-08T23:18:33Z","timestamp":1783552713738,"version":"3.55.0"},"reference-count":40,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,3,12]],"date-time":"2021-03-12T00:00:00Z","timestamp":1615507200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A nonlinear quantum boundary value problem (q-FBVP) formulated in the sense of quantum Caputo derivative, with fractional q-integro-difference conditions along with its fractional quantum-difference inclusion q-BVP are investigated in this research. To prove the solutions\u2019 existence for these quantum systems, we rely on the notions such as the condensing functions and approximate endpoint criterion (AEPC). Two numerical examples are provided to apply and validate our main results in this research work.<\/jats:p>","DOI":"10.3390\/sym13030469","type":"journal-article","created":{"date-parts":[[2021,3,15]],"date-time":"2021-03-15T02:51:48Z","timestamp":1615776708000},"page":"469","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":72,"title":["Condensing Functions and Approximate Endpoint Criterion for the Existence Analysis of Quantum Integro-Difference FBVPs"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3463-2607","authenticated-orcid":false,"given":"Shahram","family":"Rezapour","sequence":"first","affiliation":[{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 406040, Taiwan"},{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 53751-71379, Iran"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Atika","family":"Imran","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4501-9269","authenticated-orcid":false,"given":"Azhar","family":"Hussain","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3733-1239","authenticated-orcid":false,"given":"Francisco","family":"Mart\u00ednez","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Statistics, Technological University of Cartagena, 30203 Cartagena, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1574-1800","authenticated-orcid":false,"given":"Sina","family":"Etemad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 53751-71379, Iran"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2260-0341","authenticated-orcid":false,"given":"Mohammed K. 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Nonlocal Cauchy problem via a fractional operator involving power kernel in Banach spaces. Fractal Fract., 3.","DOI":"10.3390\/fractalfract3020027"},{"key":"ref_4","first-page":"181","article-title":"Economic models involving time fractal","volume":"1","author":"Golmankhane","year":"2020","journal-title":"J. Math. Model. Financ."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"7695","DOI":"10.3934\/math.2020492","article-title":"New results on complex conformable integral","volume":"5","author":"Kaabar","year":"2020","journal-title":"AIMS Math."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Yavuz, M., and Sene, N. (2020). Approximate solutions of the model describing fluid flow using generalized \u03c1-laplace transform method and heat balance integral method. Axioms, 9.","DOI":"10.3390\/axioms9040123"},{"key":"ref_7","first-page":"609","article-title":"Note on the conformable fractional derivatives and integrals of complex-valued functions of a real variable","volume":"50","author":"Kaabar","year":"2020","journal-title":"IAENG Int. J. Appl. Math."},{"key":"ref_8","first-page":"1","article-title":"Dynamical behaviors of separated homotopy method defined by conformable operator","volume":"7","author":"Yavuz","year":"2019","journal-title":"Konuralp J. Math."},{"key":"ref_9","first-page":"193","article-title":"On q-definite integrals","volume":"41","author":"Jackson","year":"1910","journal-title":"Quart. J. Pure Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1017\/S0080456800002751","article-title":"On q-functions and a certain difference operator","volume":"46","author":"Jackson","year":"1909","journal-title":"Trans. R. Soc. Edinb."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"305","DOI":"10.2307\/2370183","article-title":"q-difference equations","volume":"32","author":"Jackson","year":"1910","journal-title":"Am. J. Math."},{"key":"ref_12","first-page":"616","article-title":"q-analogue of the Cauchy formula","volume":"17","year":"1966","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1017\/S0305004100045060","article-title":"Certain fractional q-integrals and q-derivatives","volume":"66","author":"Agarwal","year":"1969","journal-title":"Proc. Camb. Philos. Soc."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Goodrich, C., and Peterson, A. (2015). Discrete Fractional Calculus, Springer.","DOI":"10.1007\/978-3-319-25562-0"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1137\/0501017","article-title":"Lie theory and q-difference equations","volume":"1","author":"Miller","year":"1970","journal-title":"Siam J. Math. Anal."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"726","DOI":"10.3906\/mat-1703-7","article-title":"An application of q-Sumudu transform for fractional q-Kinetic equation","volume":"42","author":"Purohit","year":"2018","journal-title":"Turk. J. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"311","DOI":"10.2298\/AADM0701311R","article-title":"Fractional integrals and derivatives in q-calculus","volume":"1","author":"Rajkovic","year":"2007","journal-title":"Appl. Anal. Disc. Math."},{"key":"ref_18","first-page":"119","article-title":"On the existence of solutions for fractional q-difference inclusions with q-antiperiodic boundary conditions","volume":"59","author":"Ahmad","year":"2016","journal-title":"Bull. Math. Soc. Sci. Math. Roum. Tome"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1333","DOI":"10.1007\/s00009-013-0258-0","article-title":"Existence of solutions for nonlinear fractional q-difference inclusions with nonlocal robin (separated) conditions","volume":"10","author":"Ahmad","year":"2007","journal-title":"Mediterr. J. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1186\/s13662-020-02766-y","article-title":"Solutions of two fractional q-integro-differential equations under sum and integral boundary value conditions on a time scale","volume":"2020","author":"Alzabut","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_21","first-page":"201","article-title":"Approximate solutions for a fractional q-integro-difference equation","volume":"13","author":"Balkani","year":"2019","journal-title":"J. Math. Ext."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Etemad, S., Ntouyas, S.K., and Ahmad, B. (2019). Existence theory for a fractional q-integro-difference equation with q-integral boundary conditions of different orders. Mathematics, 7.","DOI":"10.3390\/math7080659"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1186\/s13662-020-02679-w","article-title":"\u03b1-\u03c8-contractions and solutions of a q-fractional differential inclusion with three-point boundary value conditions via computational results","volume":"2020","author":"Etemad","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Ouncharoen, R., Patanarapeelert, N., and Sitthiwirattham, T. (2018). Nonlocal q-symmetric integral boundary value problem for sequential q-symmetric integro-difference equations. Mathematics, 6.","DOI":"10.3390\/math6110218"},{"key":"ref_25","first-page":"63","article-title":"Some theorems of existence of solutions for fractional hybrid q-difference inclusion","volume":"12","author":"Samei","year":"2019","journal-title":"J. Adv. Math. Stud."},{"key":"ref_26","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-integro-differential with non-separated and initial boundary conditions","volume":"2019","author":"Samei","year":"2019","journal-title":"J. Inequal. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1186\/s13662-020-02923-3","article-title":"On a fractional q-differential inclusion on a time scale via endpoints and numerical calculations","volume":"2020","author":"Samei","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Sitho, S., Sudprasert, C., Ntouyas, S.K., and Tariboon, J. (2020). Noninstantaneous impulsive fractional quantum Hahn integro-difference boundary value problems. Mathematics, 8.","DOI":"10.3390\/math8050671"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1186\/s13661-016-0522-x","article-title":"On nonlocal fractional q-integral boundary value problems of fractional q-difference and fractional q-integro-difference equations involving different numbers of order and q","volume":"2016","author":"Sitthiwirattham","year":"2016","journal-title":"Bound. Value Probl."},{"key":"ref_30","first-page":"2013","article-title":"Existence results for fractional q-difference equations with nonlocal q-integral boundary conditions","volume":"48","author":"Zhao","year":"2013","journal-title":"Adv. Differ. Equ."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"474138","DOI":"10.1155\/2014\/474138","article-title":"Existence of solutions for fractional q-integrodifference equations with nonlocal fractional q-integral conditions","volume":"2014","author":"Asawasamrit","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_32","first-page":"265","article-title":"On the existence of solutions for nonlinear fractional q-difference equations with q-integral boundary conditions","volume":"8","author":"Etemad","year":"2015","journal-title":"J. Adv. Math. Stud."},{"key":"ref_33","first-page":"2019","article-title":"Existence and uniqueness of solutions for multi-term fractional q-integro-differential equations via quantum calculus","volume":"475","author":"Ntouyas","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_34","first-page":"283","article-title":"The general theory of a class of linear partial q-difference equations","volume":"26","author":"Adams","year":"1924","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/j.camwa.2010.11.012","article-title":"Positive solutions for a class of boundary value problems with fractional q-differences","volume":"61","author":"Ferreira","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"9682","DOI":"10.1016\/j.amc.2012.03.006","article-title":"Positive solutions for a class of higher order boundary value problems with fractional q-derivatives","volume":"218","author":"Graef","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_37","first-page":"385459","article-title":"Positive solutions for boundary value problem of nonlinear fractional q-difference equation","volume":"2011","year":"2011","journal-title":"ISRN Math. Anal."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Guo, D.J., Lakshmikantham, V., and Liu, X.Z. (1996). Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers Group.","DOI":"10.1007\/978-1-4613-1281-9"},{"key":"ref_39","first-page":"1089","article-title":"Existence of solutions of initial value problems for abstract semilinear evolution equations","volume":"48","author":"Li","year":"2005","journal-title":"Acta Math. Sin. Engl. Ser. Mar."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.na.2009.06.074","article-title":"Endpoints of set-valued contractions in metric spaces","volume":"72","year":"2010","journal-title":"Nonlinear Anal."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/3\/469\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:35:01Z","timestamp":1760160901000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/3\/469"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,12]]},"references-count":40,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["sym13030469"],"URL":"https:\/\/doi.org\/10.3390\/sym13030469","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,12]]}}}