{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:46:43Z","timestamp":1760060803091,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2025,9,24]],"date-time":"2025-09-24T00:00:00Z","timestamp":1758672000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"CIDMA\u2013Center for Research and Development in Mathematics and Applications","award":["UID\/4106\/2025","UID\/PRR\/4106\/2025"],"award-info":[{"award-number":["UID\/4106\/2025","UID\/PRR\/4106\/2025"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>In this study, we investigate implicit fractional differential equations subject to anti-periodic boundary conditions. The fractional operator incorporates two distinct generalizations: the Caputo tempered fractional derivative and the Caputo fractional derivative with respect to a smooth function. We investigate the existence and uniqueness of solutions using fixed-point theorems. Stability in the sense of Ulam\u2013Hyers and Ulam\u2013Hyers\u2013Rassias is also considered. Three detailed examples are presented to illustrate the applicability and scope of the theoretical results. Several existing results in the literature can be recovered as particular cases of the framework developed in this work.<\/jats:p>","DOI":"10.3390\/math13193077","type":"journal-article","created":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T07:46:49Z","timestamp":1758786409000},"page":"3077","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Existence and Stability Analysis of Anti-Periodic Boundary Value Problems with Generalized Tempered Fractional Derivatives"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1305-2411","authenticated-orcid":false,"given":"Ricardo","family":"Almeida","sequence":"first","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3535-3909","authenticated-orcid":false,"given":"Nat\u00e1lia","family":"Martins","sequence":"additional","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"107402","DOI":"10.1016\/j.jfranklin.2024.107402","article-title":"On the stability of memory-dependent multi-agent systems under DoS attacks","volume":"362","author":"Almeida","year":"2025","journal-title":"J. Franklin Inst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"110224","DOI":"10.1016\/j.chaos.2020.110224","article-title":"Lyapunov functions for fractional-order systems in biology: Methods and applications","volume":"140","author":"Boukhouima","year":"2020","journal-title":"Chaos Solitons Fract."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Caponetto, R., Dongola, G., Fortuna, L., and Petr\u00e1\u0161, I. (2010). Fractional Order Systems: Modelling and Control Applications, World Scientific.","DOI":"10.1142\/9789814304207"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.aml.2018.06.022","article-title":"Scott-Blair models with time-varying viscosity","volume":"86","author":"Colombaro","year":"2018","journal-title":"Appl. Math. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/s40808-025-02394-z","article-title":"A fractional derivative approach to infectious disease dynamics: Modeling and optimal control strategies","volume":"11","author":"Jose","year":"2025","journal-title":"Model. Earth Syst. Environ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"467","DOI":"10.3934\/math.2020031","article-title":"Fractional physical problems including wind-influenced projectile motion with Mittag-Leffler kernel","volume":"5","author":"Ozarslan","year":"2020","journal-title":"AIMS Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1247","DOI":"10.4236\/jamp.2018.66105","article-title":"Applications of fractional calculus to Newtonian mechanics","volume":"6","author":"Varieschi","year":"2018","journal-title":"J. Appl. Math. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"7793","DOI":"10.1002\/mma.10002","article-title":"Modeling blood alcohol concentration using fractional differential equations based on the \u03c8-Caputo derivative","volume":"47","author":"Wanassi","year":"2024","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_9","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V."},{"key":"ref_10","unstructured":"Oldham, K.B., and Spanier, J. (1974). The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press."},{"key":"ref_11","unstructured":"Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993). Fractional Integrals and Derivatives, Translated from the 1987 Russian Original, Gordon and Breach."},{"key":"ref_12","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_13","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.jcp.2014.04.024","article-title":"Tempered fractional calculus","volume":"293","author":"Sabzikar","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.aml.2018.01.016","article-title":"Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation","volume":"81","author":"Sousa","year":"2018","journal-title":"Appl. Math. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"9107","DOI":"10.3934\/math.2024443","article-title":"Fractional tempered differential equations depending on arbitrary kernels","volume":"9","author":"Almeida","year":"2024","journal-title":"AIMS Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1689","DOI":"10.1007\/s11071-021-06628-4","article-title":"New theories and applications of tempered fractional differential equations","volume":"105","author":"Obeidat","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Fedorov, V.E., and Filin, N.V. (2023). A class of quasilinear equations with distributed Gerasimov-Caputo derivatives. Mathematics, 11.","DOI":"10.3390\/math11112472"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/978-3-031-28505-9_1","article-title":"Some classes of quasilinear equations with Gerasimov-Caputo derivatives","volume":"Volume 423","author":"Vasilyev","year":"2023","journal-title":"Differential Equations, Mathematical Modeling and Computational Algorithms"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1186\/s13662-018-1594-y","article-title":"Existence and uniqueness of solutions to fractional differential equations in the frame of generalized Caputo fractional derivatives","volume":"2018","author":"Gambo","year":"2018","journal-title":"Adv. Differ. Equ."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"361","DOI":"10.22436\/jmcs.034.04.04","article-title":"Existence and controllability results for neutral fractional Volterra-Fredholm integro-differential equations","volume":"34","author":"Gunasekar","year":"2024","journal-title":"J. Math. Comput. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"6844686","DOI":"10.1155\/2024\/6844686","article-title":"Existence and uniqueness of solutions for fractional-differential equation with boundary condition using nonlinear multi-fractional derivatives","volume":"2024","author":"Promsakon","year":"2024","journal-title":"Math. Probl. Eng."},{"key":"ref_22","first-page":"595","article-title":"Nonlocal solvability of quasilinear degenerate equations with Gerasimov-Caputo derivatives","volume":"44","author":"Yadrikhinskiy","year":"2023","journal-title":"Lobachevskii J. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"962","DOI":"10.1007\/s13540-022-00049-9","article-title":"Stability analysis of fractional differential equations with the short-term memory property","volume":"25","author":"Hai","year":"2022","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"6538","DOI":"10.1016\/j.jfranklin.2018.12.033","article-title":"Stability analysis of nonlinear Hadamard fractional differential system","volume":"356","author":"Wang","year":"2019","journal-title":"J. Frankl. Inst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"9894","DOI":"10.3934\/math.2022552","article-title":"Some qualitative properties of solutions to a nonlinear fractional differential equation involving two Caputo fractional derivatives","volume":"7","author":"Derbazi","year":"2022","journal-title":"AIMS Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1186\/s13660-025-03261-2","article-title":"A compact finite difference scheme for solving fractional Black-Scholes option pricing model","volume":"2025","author":"Feng","year":"2025","journal-title":"J. Inequal. Appl."},{"key":"ref_27","first-page":"127655","article-title":"Second-order accurate, robust and efficient ADI Galerkin technique for the three-dimensional nonlocal heat model arising in viscoelasticity","volume":"440","author":"Luo","year":"2023","journal-title":"Appl. Math Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.matcom.2021.10.010","article-title":"Finite difference method for solving fractional differential equations at irregular meshes","volume":"193","author":"Vargas","year":"2022","journal-title":"Math. Comput. Simul."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1186\/s13662-022-03697-6","article-title":"Numerical solution of fractional differential equations with Caputo derivative by using numerical fractional predict\u2013correct technique","volume":"2022","author":"Zabidi","year":"2022","journal-title":"Adv. Cont. Discr. Mod."},{"key":"ref_30","first-page":"295","article-title":"Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory","volume":"35","author":"Ahmad","year":"2010","journal-title":"Topol. Methods Nonlinear Anal."},{"key":"ref_31","first-page":"457","article-title":"Existence of solutions for impulsive anti-periodic boundary value problems of fractional semilinear evolution equations","volume":"18","author":"Agarwal","year":"2011","journal-title":"Dyna. Contin. Discrete Impuls. Syst. Ser. A Math. Anal."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Almeida, R. (2025). A unified approach to implicit fractional differential equations with anti-periodic boundary conditions. Mathematics, 13.","DOI":"10.3390\/math13172890"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1186\/s13662-019-2077-5","article-title":"Existence and Ulam-Hyers stability for Caputo conformable differential equations with four-point integral conditions","volume":"2019","author":"Aphithana","year":"2019","journal-title":"Adv. Differ. Equ."}],"container-title":["Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-7390\/13\/19\/3077\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:49:13Z","timestamp":1760035753000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-7390\/13\/19\/3077"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,24]]},"references-count":33,"journal-issue":{"issue":"19","published-online":{"date-parts":[[2025,10]]}},"alternative-id":["math13193077"],"URL":"https:\/\/doi.org\/10.3390\/math13193077","relation":{},"ISSN":["2227-7390"],"issn-type":[{"type":"electronic","value":"2227-7390"}],"subject":[],"published":{"date-parts":[[2025,9,24]]}}}