{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:46:39Z","timestamp":1760060799410,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,19]],"date-time":"2025-09-19T00:00:00Z","timestamp":1758240000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Research Project of Anhui Educational Committee","award":["2024AH051679"],"award-info":[{"award-number":["2024AH051679"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The main objective of this work is to discuss the generalized anti-periodic boundary conditions of the generalized Caputo fractional differential equations with p(t)-Laplacian operators. By applying the Schaefer fixed point theorem, the existence of mild solutions to this problem is obtained, which generalizes and enriches the anti-periodic boundary value problem of Caputo fractional hybrid differential equations. Finally, a numerical example is given to verify our main results. The anti-periodic boundary value condition imparts a form of symmetric inversion with respect to the original state, so it exhibits an anti-symmetric structural feature.<\/jats:p>","DOI":"10.3390\/sym17091569","type":"journal-article","created":{"date-parts":[[2025,9,19]],"date-time":"2025-09-19T10:08:58Z","timestamp":1758276538000},"page":"1569","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Existence of Mild Solutions for the Generalized Anti-Periodic Boundary Value Problem to the Fractional Hybird Differential Equations with p(t)-Laplacian Operator"],"prefix":"10.3390","volume":"17","author":[{"given":"Jinxiu","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Huaibei Normal University, Huaibei 235000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guanghao","family":"Jiang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Huaibei Normal University, Huaibei 235000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tengfei","family":"Shen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Huaibei Normal University, Huaibei 235000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"5547804","DOI":"10.1155\/2021\/5547804","article-title":"A Semianalytical approach to the solution of time-fractional Navier-Stokes equation","volume":"2021","author":"Ali","year":"2021","journal-title":"Adv. Math. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1140\/epjp\/s13360-022-02603-z","article-title":"A fractional-order mathematical model for COVID-19 outbreak with the effect of symptomatic and asymptomatic transmissions","volume":"137","author":"Ali","year":"2022","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Xu, K., Chen, L., Lopes, A.M., Wang, M., Wu, R., and Zhu, M. (2023). Fractional-order Zener model with temperature-order equivalence for viscoelastic dampers. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7100714"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"552","DOI":"10.1186\/s13662-020-03005-0","article-title":"Oblique explicit wave solutions of the fractional biological population (BP) and equal width (EW) models","volume":"2020","author":"Khater","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_5","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Zhou, Y. (2023). Basic Theory of Fractional Differential Equations, World Scientific.","DOI":"10.1142\/13289"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1631","DOI":"10.1016\/S0252-9602(16)30095-9","article-title":"A nonlocal hybrid boundary value problem of Caputo fractional integro-differential equations","volume":"36","author":"Ahmad","year":"2016","journal-title":"Acta Math. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1016\/j.chaos.2018.07.009","article-title":"Fractional Langevin equation with anti-periodic boundary conditions","volume":"114","author":"Fazli","year":"2018","journal-title":"Chaos Solitons Fractals"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3019","DOI":"10.1016\/j.aej.2020.04.053","article-title":"On a fractional hybrid integro-differential equation with mixed hybrid integral boundary value conditions by using three operators","volume":"59","author":"Baleanu","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1186\/s13661-020-01332-5","article-title":"Existence of solutions for integral boundary value problems of mixed fractional differential equations under resonance","volume":"2020","author":"Song","year":"2020","journal-title":"Bound. Value Probl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3231","DOI":"10.1080\/00036811.2020.1839645","article-title":"Solvability of Langevin equations with two Hadamard fractional derivatives via Mittag-Leffler functions","volume":"101","author":"Abbas","year":"2022","journal-title":"Appl. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/S0022-247X(02)00376-1","article-title":"A Knobloch-type result for p(t)-Laplacian systems","volume":"282","author":"Fan","year":"2003","journal-title":"J. Math. Anal. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13661-017-0900-z","article-title":"Existence of solutions for fractional Sturm-Liouville boundary value problems with p(t)-Laplacian operator","volume":"2017","author":"Xue","year":"2017","journal-title":"Bound. Value Probl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1007\/s12190-019-01264-z","article-title":"The existence of solutions for mixed fractional resonant boundary value problem with p(t)-Laplacian operator","volume":"61","author":"Tang","year":"2019","journal-title":"J. Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"231","DOI":"10.2298\/FIL2201231G","article-title":"Existence of Solutions for Weighted p(t)-Laplacian Mixed Caputo Fractional Dierential Equations at Resonance","volume":"36","author":"Lakoud","year":"2022","journal-title":"Filomat"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4421","DOI":"10.3934\/mbe.2023205","article-title":"A periodic boundary value problem of fractional differential equation involving p(t)-Laplacian operator","volume":"20","author":"Xue","year":"2023","journal-title":"Math. Biosci. Eng"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"20240045","DOI":"10.1515\/dema-2024-0045","article-title":"Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator","volume":"57","author":"Sun","year":"2024","journal-title":"Demonstr. Math."},{"key":"ref_18","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_19","doi-asserted-by":"crossref","unstructured":"Almeida, R. (2020). Functional differential equations involving the \u03c8-Caputo fractional derivative. Fractal Fract., 4.","DOI":"10.3390\/fractalfract4020029"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1186\/s13661-024-01872-0","article-title":"On qualitative analysis of a fractional hybrid Langevin differential equation with novel boundary conditions","volume":"2024","author":"Ali","year":"2024","journal-title":"Bound. Value Probl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3458858","DOI":"10.1155\/2023\/3458858","article-title":"On the Existence and Stability of Positive Solutions of Eigenvalue Problems for a Class of p-Laplacian \u03c8-Caputo Fractional Integro-Differential Equations","volume":"2023","author":"Awad","year":"2023","journal-title":"J. Math."},{"key":"ref_22","first-page":"4991","article-title":"Reliable computational method for systems of fractional differential equations endowed with \u03c8-Caputo fractional derivative","volume":"5","author":"Alyami","year":"2024","journal-title":"Contemp. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"9456","DOI":"10.1002\/mma.10810","article-title":"Existence of Solutions for a Coupled System of \u03c8-Caputo Fractional Differential Equations with Integral Boundary Conditions","volume":"48","author":"Poovarasan","year":"2025","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1671","DOI":"10.1016\/j.aml.2012.01.035","article-title":"An anti-periodic boundary value problem for the fractional differential equation with a p-Laplacian operator","volume":"25","author":"Chen","year":"2012","journal-title":"Appl. Math. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.jmaa.2006.03.087","article-title":"Existence of solutions for weighted p(r)-Laplacian system boundary value problems","volume":"327","author":"Zhang","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1007\/BF01362380","article-title":"\u00dcber die Methode der a priori-Schranken","volume":"129","author":"Schaefer","year":"1955","journal-title":"Math. Ann."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Schaefer, H.H., and Wolff, M.P. (1999). Topological Vector Spaces, Springer. 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