{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T21:44:17Z","timestamp":1773956657922,"version":"3.50.1"},"reference-count":23,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,7]],"date-time":"2025-09-07T00:00:00Z","timestamp":1757203200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Mongkut\u2019s University of Technology North Bangkok","award":["KMUTNB-67-KNOW-16"],"award-info":[{"award-number":["KMUTNB-67-KNOW-16"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper is concerned with the existence and uniqueness of solutions for a coupled system of (k,\u03c8)-Hilfer and (k,\u03c8)-Caputo sequential fractional differential equations with non-separated boundary conditions. We make use of the Banach contraction mapping principle to obtain the uniqueness result, while two existence results are proved by using Leray\u2013Schauder nonlinear alternative and Krasnosel\u2019ski\u012d\u2019s fixed point theorem. The obtained results are illustrated by numerical examples.<\/jats:p>","DOI":"10.3390\/axioms14090685","type":"journal-article","created":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T09:32:01Z","timestamp":1757496721000},"page":"685","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Coupled System of (k, \u03c8)-Hilfer and (k, \u03c8)-Caputo Sequential Fractional Differential Equations with Non-Separated Boundary Conditions"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1290-9302","authenticated-orcid":false,"given":"Furkan","family":"Erkan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Ege University, Bornova 35100, Izmir, T\u00fcrkiye"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7393-2819","authenticated-orcid":false,"given":"Nuket Aykut","family":"Hamal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Ege University, Bornova 35100, Izmir, T\u00fcrkiye"}]},{"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, 451 10 Ioannina, Greece"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8185-3539","authenticated-orcid":false,"given":"Jessada","family":"Tariboon","sequence":"additional","affiliation":[{"name":"Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-5251-3310","authenticated-orcid":false,"given":"Phollakrit","family":"Wongsantisuk","sequence":"additional","affiliation":[{"name":"Department of Electronics Engineering Technology, College of Industrial Technology, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,7]]},"reference":[{"key":"ref_1","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier Science. North-Holland Mathematics Studies."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2000). Applications of Fractional Calculus in Physics, World Scientific.","DOI":"10.1142\/9789812817747"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"106755","DOI":"10.1016\/j.aml.2020.106755","article-title":"The averaging principle of Hilfer fractional stochastic delay differential equations with Poisson jump","volume":"112","author":"Ahmed","year":"2021","journal-title":"Appl. Math. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"107549","DOI":"10.1016\/j.aml.2021.107549","article-title":"A novel result on averaging principle of stochastic Hilfer-type fractional system involving non-Lipschitz coefficients","volume":"112","author":"Luo","year":"2021","journal-title":"Appl. Math. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1254","DOI":"10.1080\/16583655.2022.2157188","article-title":"Controllability of damped dynamical systems modelled by Hilfer fractional derivatives","volume":"16","author":"Naveen","year":"2022","journal-title":"J. Taibah Univ. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2891","DOI":"10.1016\/j.aej.2020.01.055","article-title":"New results on nonlocal functional integro-differential equations via Hilfer fractional derivative","volume":"59","author":"Subashini","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1016\/j.cnsns.2016.09.006","article-title":"A Caputo fractional derivative of a function with respect to another function","volume":"44","author":"Almeida","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.cnsns.2018.01.005","article-title":"On the \u03c8-Hilfer fractional derivative","volume":"60","author":"Sousa","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_9","first-page":"89","article-title":"k-Fractional integrals and application","volume":"7","author":"Mubeen","year":"2012","journal-title":"Int. J. Contemp. Math. Sci."},{"key":"ref_10","first-page":"179","article-title":"On hypergeometric functions and Pochhammer k-symbol","volume":"15","author":"Diaz","year":"2004","journal-title":"Divulg. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"41","DOI":"10.12988\/ijcms.2013.13004","article-title":"On the k-Riemann\u2013Liouville fractional derivative","volume":"8","author":"Romero","year":"2013","journal-title":"Int. J. Contemp. Math. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"64946","DOI":"10.1109\/ACCESS.2018.2878266","article-title":"Generalized Riemann-Liouville k-fractional integrals associated with Ostrowski type inequalities and error bounds of Hadamard inequalities","volume":"6","author":"Kwun","year":"2018","journal-title":"IEEE Access"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"111335","DOI":"10.1016\/j.chaos.2021.111335","article-title":"On the nonlinear (k, \u03c8)-Hilfer fractional differential equations","volume":"152","author":"Kucche","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Ntouyas, S.K., Ahmad, B., and Tariboon, J. (2022). (k, \u03c8)-Hilfer nonlocal integro-multi-point boundary value problems for fractional differential equations and inclusions. Mathematics, 10.","DOI":"10.3390\/math10152615"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Samadi, A., Ntouyas, S.K., and Tariboon, J. (2022). Nonlocal coupled system for (k, \u03c8)-Hilfer fractional differential equations. Fractal Fract., 5.","DOI":"10.3390\/fractalfract6050234"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"103111","DOI":"10.1016\/j.asej.2024.103111","article-title":"On the (k, \u03c6)-Hilfer Langevin fractional coupled system having multi point boundary conditions and fractional integrals","volume":"15","author":"Pan","year":"2024","journal-title":"Ain Shams Eng. J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1007\/s41808-022-00173-w","article-title":"(k, \u03c8)-Hilfer variational problem","volume":"8","author":"Torres","year":"2022","journal-title":"J. Elliptic Parabol. Equ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3033","DOI":"10.1007\/s12190-024-02078-4","article-title":"Controllability of fractional dynamical systems with (k, \u03c8)-Hilfer fractional derivative","volume":"70","author":"Haque","year":"2024","journal-title":"J. Appl. Math. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Samadi, A., Ntouyas, S.K., and Tariboon, J. (2024). Mixed Hilfer and Caputo fractional Riemann\u2013Stieltjes integro-differential equations with non-separated boundary conditions. Mathematics, 12.","DOI":"10.3390\/math12091361"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Samadi, A., Ntouyas, S.K., and Tariboon, J. (2024). Fractional sequential coupled systems of Hilfer and Caputo integro-differential equations with non-separated boundary conditions. Axioms, 13.","DOI":"10.3390\/axioms13070484"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Erkan, F., Aykut Hamal, N., Ntouyas, S.K., and Tariboon, J. (2025). Existence and uniqueness analysis for (k, \u03c8)-Hilfer and (k, \u03c8)-Caputo sequential fractional differential equations and inclusions with non-separated boundary conditions. Fractal Fract., 9.","DOI":"10.3390\/fractalfract9070437"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Granas, A., and Dugundji, J. (2003). Fixed Point Theory, Springer.","DOI":"10.1007\/978-0-387-21593-8"},{"key":"ref_23","unstructured":"Smart, D.R. (1980). Fixed Point Theory, Cambridge University Press."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/9\/685\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:41:30Z","timestamp":1760035290000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/9\/685"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,7]]},"references-count":23,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["axioms14090685"],"URL":"https:\/\/doi.org\/10.3390\/axioms14090685","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,7]]}}}