{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,18]],"date-time":"2026-06-18T20:45:35Z","timestamp":1781815535807,"version":"3.54.5"},"reference-count":27,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,10,29]],"date-time":"2022-10-29T00:00:00Z","timestamp":1667001600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia","award":["793"],"award-info":[{"award-number":["793"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this study, we examine the existence and Hyers\u2013Ulam stability of a coupled system of generalized Liouville\u2013Caputo fractional order differential equations with integral boundary conditions and a connection to Katugampola integrals. In the first and third theorems, the Leray\u2013Schauder alternative and Krasnoselskii\u2019s fixed point theorem are used to demonstrate the existence of a solution. The Banach fixed point theorem\u2019s concept of contraction mapping is used in the second theorem to emphasise the analysis of uniqueness, and the results for Hyers\u2013Ulam stability are established in the next theorem. We establish the stability of Ulam\u2013Hyers using conventional functional analysis. Finally, examples are used to support the results. When a generalized Liouville\u2013Caputo (\u03c1) parameter is modified, asymmetric results are obtained. This study presents novel results that significantly contribute to the literature on this topic.<\/jats:p>","DOI":"10.3390\/sym14112273","type":"journal-article","created":{"date-parts":[[2022,10,30]],"date-time":"2022-10-30T10:47:57Z","timestamp":1667126877000},"page":"2273","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["On the Generalized Liouville\u2013Caputo Type Fractional Differential Equations Supplemented with Katugampola Integral Boundary Conditions"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6447-6361","authenticated-orcid":false,"given":"Muath","family":"Awadalla","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, King Faisal University, Hafuf, Al Ahsa 31982, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5281-0935","authenticated-orcid":false,"given":"Muthaiah","family":"Subramanian","sequence":"additional","affiliation":[{"name":"Department of Mathematics, KPR Institute of Engineering and Technology, Coimbatore 641407, Tamil Nadu, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2744-6320","authenticated-orcid":false,"given":"Kinda","family":"Abuasbeh","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, King Faisal University, Hafuf, Al Ahsa 31982, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7975-7545","authenticated-orcid":false,"given":"Murugesan","family":"Manigandan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore 641020, Tamil Nadu, India"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Britton, N.F. (2003). Essential Mathematical Biology, Springer.","DOI":"10.1007\/978-1-4471-0049-2"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Ma, Y., and Ji, D. (2022). Existence of Solutions to a System of Riemann-Liouville Fractional Differential Equations with Coupled Riemann-Stieltjes Integrals Boundary Conditions. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6100543"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Theswan, S., Ntouyas, S.K., Ahmad, B., and Tariboon, J. (2022). Existence Results for Nonlinear Coupled Hilfer Fractional Differential Equations with Nonlocal Riemann\u2013Liouville and Hadamard-Type Iterated Integral Boundary Conditions. Symmetry, 14.","DOI":"10.3390\/sym14091948"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Klafter, J., Lim, S., and Metzler, R. (2012). Fractional Dynamics: Recent Advances, World Scientific.","DOI":"10.1142\/9789814340595"},{"key":"ref_5","unstructured":"Podlubny, I. (1998). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"552","DOI":"10.2478\/s13540-014-0185-1","article-title":"Some pioneers of the applications of fractional calculus","volume":"17","author":"Valerio","year":"2014","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1140","DOI":"10.1016\/j.cnsns.2010.05.027","article-title":"Recent history of fractional calculus","volume":"16","author":"Machado","year":"2011","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_8","unstructured":"Kilbas, A.A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier Science Limited, North-Holland Mathematics Studies."},{"key":"ref_9","first-page":"398","article-title":"On some simple generalizations of linear elliptic boundary problems","volume":"10","author":"Bitsadze","year":"1969","journal-title":"Soviet Math. Dokl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"253","DOI":"10.15388\/NA.17.3.14054","article-title":"Numerical approximation of one model of bacterial self-organization","volume":"17","author":"Ciegis","year":"2012","journal-title":"Nonlinear Anal. Model. Control."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-021-03414-9","article-title":"Existence, uniqueness and stability analysis of a coupled fractional-order differential systems involving hadamard derivatives and associated with multi-point boundary conditions","volume":"2021","author":"Subramanian","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Rahmani, A., Du, W.S., Khalladi, M.T., Kosti\u0107, M., and Velinov, D. (2022). Proportional Caputo Fractional Differential Inclusions in Banach Spaces. Symmetry, 14.","DOI":"10.3390\/sym14091941"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Tudorache, A., and Luca, R. (2022). Positive Solutions for a Fractional Differential Equation with Sequential Derivatives and Nonlocal Boundary Conditions. Symmetry, 14.","DOI":"10.3390\/sym14091779"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"107018","DOI":"10.1016\/j.aml.2021.107018","article-title":"Existence and uniqueness results for a nonlinear coupled system involving caputo fractional derivatives with a new kind of coupled boundary conditions","volume":"116","author":"Ahmad","year":"2021","journal-title":"Appl. Math. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Alsaedi, A., Alghanmi, M., Ahmad, B., and Ntouyas, S.K. (2018). Generalized liouville\u2013caputo fractional differential equations and inclusions with nonlocal generalized fractional integral and multipoint boundary conditions. Symmetry, 10.","DOI":"10.3390\/sym10120667"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-021-03525-3","article-title":"On a nonlinear sequential four-point fractional q-difference equation involving q-integral operators in boundary conditions along with stability criteria","volume":"2021","author":"Boutiara","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-020-02690-1","article-title":"A coupled system of generalized sturm\u2013liouville problems and langevin fractional differential equations in the framework of nonlocal and nonsingular derivatives","volume":"2020","author":"Baleanu","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Muthaiah, S., and Baleanu, D. (2020). Existence of solutions for nonlinear fractional differential equations and inclusions depending on lower-order fractional derivatives. Axioms, 9.","DOI":"10.3390\/axioms9020044"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Saeed, A.M., Abdo, M.S., and Jeelani, M.B. (2021). Existence and Ulam\u2013Hyers stability of a fractional-order coupled system in the frame of generalized Hilfer derivatives. Mathematics, 9.","DOI":"10.3390\/math9202543"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Ahmad, D., Agarwal, R.P., and Rahman, G.U.R. (2022). Formulation, Solution\u2019s Existence, and Stability Analysis for Multi-Term System of Fractional-Order Differential Equations. Symmetry, 14.","DOI":"10.3390\/sym14071342"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Samadi, A., Ntouyas, S.K., and Tariboon, J. (2022). On a nonlocal coupled system of Hilfer generalized proportional fractional differential equations. Symmetry, 14.","DOI":"10.3390\/sym14040738"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Awadalla, M., Abuasbeh, K., Subramanian, M., and Manigandan, M. (2022). On a System of \u03c8-Caputo Hybrid Fractional Differential Equations with Dirichlet Boundary Conditions. Mathematics, 10.","DOI":"10.3390\/math10101681"},{"key":"ref_23","first-page":"860","article-title":"New approach to a generalized fractional integral","volume":"218","author":"Katugampola","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Katugampola, U.N. (2011). A new approach to generalized fractional derivatives. arXiv.","DOI":"10.1016\/j.amc.2011.03.062"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2607","DOI":"10.22436\/jnsa.010.05.27","article-title":"On the generalized fractional derivatives and their caputo modification","volume":"10","author":"Jarad","year":"2017","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_26","unstructured":"Granas, A., and Dugundji, J. (2013). Fixed Point Theory, Springer Science & Business Media."},{"key":"ref_27","first-page":"123","article-title":"Two remarks on the method of successive approximations, uspehi mat","volume":"10","year":"1955","journal-title":"Nauk"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2273\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:05:45Z","timestamp":1760144745000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2273"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,29]]},"references-count":27,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["sym14112273"],"URL":"https:\/\/doi.org\/10.3390\/sym14112273","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,29]]}}}