{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:33:38Z","timestamp":1760236418108,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,11,19]],"date-time":"2021-11-19T00:00:00Z","timestamp":1637280000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Science, Research and Innovation Promotion Fund  under Basic Research Plan - Saun Dusit University","award":["-"],"award-info":[{"award-number":["-"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we study a boundary value problem involving (p,q)-integrodifference equations, supplemented with nonlocal fractional (p,q)-integral boundary conditions with respect to asymmetric operators. First, we convert the given nonlinear problem into a fixed-point problem, by considering a linear variant of the problem at hand. Once the fixed-point operator is available, existence and uniqueness results are established using the classical Banach\u2019s and Schaefer\u2019s fixed-point theorems. The application of the main results is demonstrated by presenting numerical examples. Moreover, we study some properties of (p,q)-integral that are used in our study.<\/jats:p>","DOI":"10.3390\/sym13112212","type":"journal-article","created":{"date-parts":[[2021,11,19]],"date-time":"2021-11-19T08:29:17Z","timestamp":1637310557000},"page":"2212","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Separate Fractional (p,q)-Integrodifference Equations via Nonlocal Fractional (p,q)-Integral Boundary Conditions"],"prefix":"10.3390","volume":"13","author":[{"given":"Thongchai","family":"Dumrongpokaphan","sequence":"first","affiliation":[{"name":"Research Group in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7695-2118","authenticated-orcid":false,"given":"Sotiris K.","family":"Ntouyas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece"},{"name":"Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8455-1402","authenticated-orcid":false,"given":"Thanin","family":"Sitthiwirattham","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"305","DOI":"10.2307\/2370183","article-title":"On q-difference equations","volume":"32","author":"Jackson","year":"1910","journal-title":"Am. J. Math."},{"key":"ref_2","first-page":"193","article-title":"On q-difference integrals","volume":"41","author":"Jackson","year":"1910","journal-title":"Q. J. Pure Appl. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5683","DOI":"10.1088\/0305-4470\/24\/24\/007","article-title":"A (p,q)-oscillator realization of two-parameter quantum algebras","volume":"24","author":"Chakrabarti","year":"1991","journal-title":"J. Math. Phys. Math. Gen."},{"key":"ref_4","first-page":"392","article-title":"Some approximation results by (p,q)-analogue of Bernstein-Stancu operators","volume":"264","author":"Mursaleen","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"639","DOI":"10.2298\/FIL1603639M","article-title":"Some approximation results on Bleimann-Butzer-Hahn operators defined by (p,q)-integers","volume":"30","author":"Mursaleen","year":"2016","journal-title":"Filomat"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1725","DOI":"10.1007\/s11785-016-0553-4","article-title":"Approximation by (p,q)-Lorentz polynomials on a compact disk","volume":"10","author":"Mursaleen","year":"2016","journal-title":"Complex Anal. Oper. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4351","DOI":"10.2298\/FIL1812351R","article-title":"Convergence of iterates of q-Bernstein and (p,q)-Bernstein operators and the Kelisky-Rivlin type theorem","volume":"32","author":"Rahman","year":"2018","journal-title":"Filomat"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"458","DOI":"10.1016\/j.cam.2016.12.016","article-title":"B\u00e9zier curves based on Lupa\u015f (p,q)-analogue of Bernstein functions in CAGD","volume":"317","author":"Khan","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"ref_9","unstructured":"Jagannathan, R., and Rao, K.S. (2006). Two-parameter quantum algebras, twin-basic number, and associated generalized hypergeometric series. arXiv."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1016\/j.jmaa.2007.01.072","article-title":"Representations of two parameter quantum algebras and (p,q)-special functions","volume":"335","author":"Sahai","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_11","unstructured":"Sadjang, P.N. (2018). On the fundamental theorem of (p,q)-calculus and some (p,q)-Taylor formulas. arXiv."},{"key":"ref_12","unstructured":"Duran, U. (2016). Post Quantum Calculus. [Master\u2019s Thesis, University of Gaziantep]."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1007\/s40590-016-0139-1","article-title":"(p,q)-Beta functions and applications in approximation","volume":"24","author":"Milovanovic","year":"2018","journal-title":"Bol. Soc. Mat. Mex."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1186\/s13660-019-2053-3","article-title":"(p,q)-gamma operators which preserve x2","volume":"2019","author":"Cheng","year":"2019","journal-title":"J. Inequal. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1186\/s13662-020-2512-7","article-title":"On fractional (p,q)-Calculus","volume":"2020","author":"Soontharanon","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_16","first-page":"504","article-title":"Fractional (p,q)-calculus on finite intervals and some Integral inequalities","volume":"3","author":"Pheak","year":"2021","journal-title":"Symmetry"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1186\/s13662-021-03487-6","article-title":"Praveen Agarwal, Some trapezoid and midpoint type inequalities via fractional (p,q)-calculus","volume":"2021","author":"Pheak","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"9089865","DOI":"10.1155\/2018\/9089865","article-title":"On the second-order (p,q)-difference equations with separated boundary conditions","volume":"2018","author":"Promsakon","year":"2018","journal-title":"Adv. Math. Phys."},{"key":"ref_19","first-page":"183","article-title":"Nonlocal boundary value problems for (p,q)-difference equations","volume":"10","author":"Kamsrisuk","year":"2018","journal-title":"Differ. Equ. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1186\/s13662-020-02555-7","article-title":"Impulsive quantum (p,q)-difference equations","volume":"2020","author":"Nuntigrangjana","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_21","unstructured":"Qin, Z., and Sun, S. (2021). Positive solutions for fractional (p,q)-difference boundary value problems. J. Appl. Math. Comput., 1\u20138."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Qin, Z., and Sun, S. (2021). On a nonlinear fractional (p,q)-difference Schr\u00f6dinger equation. J. Appl. Math. Comput.","DOI":"10.1007\/s12190-021-01586-x"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1186\/s13662-020-02806-7","article-title":"Existence results of nonlocal Robin boundary value problems for fractional (p,q)-integrodifference equations","volume":"2020","author":"Soontharanon","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_24","unstructured":"Smart, D.R. (1980). Fixed Point Theorems, Cambridge University Press."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2212\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:32:52Z","timestamp":1760167972000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2212"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,19]]},"references-count":24,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2021,11]]}},"alternative-id":["sym13112212"],"URL":"https:\/\/doi.org\/10.3390\/sym13112212","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,11,19]]}}}