{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:11:46Z","timestamp":1760145106746,"version":"build-2065373602"},"reference-count":48,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,20]],"date-time":"2024-06-20T00:00:00Z","timestamp":1718841600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Applied Technology College of Soochow University"},{"name":"Taizhou University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The Hadamard fractional derivative and integral are important parts of fractional calculus which have been widely used in engineering, biology, neural networks, control theory, and so on. In addition, the periodic boundary conditions are an important class of symmetric two-point boundary conditions for differential equations and have wide applications. Therefore, this article considers a class of nonlinear Hadamard fractional coupling (p1,p2)-Laplacian systems with periodic boundary value conditions. Based on nonlinear analysis methods and the contraction mapping principle, we obtain some new and easily verifiable sufficient criteria for the existence and uniqueness of solutions to this system. Moreover, we further discuss the generalized Ulam\u2013Hyers (GUH) stability of this problem by using some inequality techniques. Finally, three examples and simulations explain the correctness and availability of our main results.<\/jats:p>","DOI":"10.3390\/sym16060774","type":"journal-article","created":{"date-parts":[[2024,6,21]],"date-time":"2024-06-21T05:33:28Z","timestamp":1718948008000},"page":"774","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Stability and Numerical Simulation of a Nonlinear Hadamard Fractional Coupling Laplacian System with Symmetric Periodic Boundary Conditions"],"prefix":"10.3390","volume":"16","author":[{"given":"Xiaojun","family":"Lv","sequence":"first","affiliation":[{"name":"Applied Technology College, Soochow University, Suzhou 215325, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2236-3016","authenticated-orcid":false,"given":"Kaihong","family":"Zhao","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Electronics & Information Engineering, Taizhou University, Taizhou 318000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haiping","family":"Xie","sequence":"additional","affiliation":[{"name":"Applied Technology College, Soochow University, Suzhou 215325, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,20]]},"reference":[{"key":"ref_1","first-page":"101","article-title":"Essai sur l\u2019\u00e9tude des fonctions donn\u00e9es par leur d\u00e9veloppment de Taylor","volume":"8","author":"Hadamard","year":"1892","journal-title":"J. Math. Pures Appl."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Ahmad, B., Alsaedi, A., Ntouyas, S., and Tariboon, J.J. (2017). Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Springer.","DOI":"10.1007\/978-3-319-52141-1"},{"key":"ref_3","unstructured":"Kilbas, A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Baleanu, D., Diethelm, K., Scalas, E., and Trujillo, J.J. (2012). Fractional Calculus: Models and Numerical Methods, World Scientific.","DOI":"10.1142\/9789814355216"},{"key":"ref_5","unstructured":"Miller, K., and Ross, B. (1993). An introduction to the Fractional Calculus and Differential Equations, Wiley."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Zhou, Y. (2014). Basic Theory of Fractional Differential Equations, World Scientific.","DOI":"10.1142\/9069"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.chaos.2016.05.005","article-title":"A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions","volume":"91","author":"Aljoudi","year":"2016","journal-title":"Chaos Soliton. Fract."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1007\/s00009-017-1012-9","article-title":"Boundary value problems for nonlinear implicit Caputo-Hadamard-type fractional differential equations with impulses","volume":"14","author":"Benchohra","year":"2017","journal-title":"Mediterr. J. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"19221","DOI":"10.3934\/math.20221055","article-title":"On solvability of BVP for a coupled Hadamard fractional systems involving fractional derivative impulses","volume":"7","author":"Huang","year":"2022","journal-title":"AIMS Math."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Mouy, M., Boulares, H., Alshammari, S., Alshammari, M., Laskri, Y., and Mohammed, W.W. (2023). On averaging principle for Caputo-Hadamard fractional stochastic differential pantograph equation. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7010031"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Ortigueira, M., and Bohannan, G. (2023). Fractional scale calculus: Hadamard vs. Liouville. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7040296"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.2298\/FIL2304053Z","article-title":"Existence and UH-stability of integral boundary problem for a class of nonlinear higher-order Hadamard fractional Langevin equation via Mittag\u2013Leffler functions","volume":"37","author":"Zhao","year":"2023","journal-title":"Filomat"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/s12346-024-01044-6","article-title":"Hadamard fractional differential equations on an unbounded domain with integro-initial conditions","volume":"23","author":"Nyamoradi","year":"2024","journal-title":"Qual. Theor. Dyn. Syst."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Alruwaily, Y., Venkatachalam, K., and El-hady, E. (2024). On some impulsive fractional integro-differential equation with anti-periodic conditions. Fractal Fract., 8.","DOI":"10.3390\/fractalfract8040219"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"025207","DOI":"10.1088\/1402-4896\/ad185b","article-title":"Integro-differential equations implicated with Caputo-Hadamard derivatives under nonlocal boundary constraints","volume":"99","author":"Hammad","year":"2024","journal-title":"Phys. Scr."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1007\/s13540-023-00235-3","article-title":"Convergence to logarithmic-type functions of solutions of fractional systems with Caputo-Hadamard and Hadamard fractional derivatives","volume":"27","author":"Kassim","year":"2024","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"20220226","DOI":"10.1515\/dema-2022-0226","article-title":"Solvability for a system of Hadamard-type hybrid fractional differential inclusions","volume":"56","author":"Zhang","year":"2023","journal-title":"Demonstr. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"8945","DOI":"10.1002\/mma.9028","article-title":"Analysis of p-Laplacian Hadamard fractional boundary value problems with the derivative term involved in the nonlinear term","volume":"46","author":"Ciftci","year":"2023","journal-title":"Math. Method. Appl. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1016\/j.aej.2023.11.081","article-title":"On Caputo-Hadamard fractional pantograph problem of two different orders with Dirichlet boundary conditions","volume":"86","author":"Rafeeq","year":"2024","journal-title":"Alex. Eng. J."},{"key":"ref_20","first-page":"7","article-title":"General problem of the movement of a compressible uid in a porous medium","volume":"9","author":"Leibenson","year":"1983","journal-title":"Izv. Akad. Nauk Kirg. SSR Ser. Biol. Nauk"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Zhao, K. (2023). Solvability, Approximation and Stability of Periodic Boundary Value Problem for a Nonlinear Hadamard Fractional Differential Equation with p-Laplacian. Axioms, 12.","DOI":"10.3390\/axioms12080733"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1186\/s13662-024-03801-y","article-title":"Study on the stability and its simulation algorithm of a nonlinear impulsive ABC-fractional coupled system with a Laplacian operator via F-contractive mapping","volume":"2024","author":"Zhao","year":"2024","journal-title":"Adv. Contin. Discret. Models"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"14767","DOI":"10.3934\/math.2023755","article-title":"Multiple positive solutions for system of mixed Hadamard fractional boundary value problems with (p1,p2)-Laplacian operator","volume":"8","author":"Rao","year":"2023","journal-title":"AIMS Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"14187","DOI":"10.3934\/math.2022782","article-title":"A new study on the existence and stability to a system of coupled higher-order nonlinear BVP of hybrid FDEs under the p-Laplacian operator","volume":"7","author":"Alkhazzan","year":"2022","journal-title":"AIMS Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3430","DOI":"10.1002\/mma.4835","article-title":"Analysis of positive solution and Hyers-Ulam stability for a class of singular fractional differential equations with p-Laplacian in Banach space","volume":"41","author":"Khan","year":"2018","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1016\/j.apnum.2021.03.001","article-title":"Multiple positive solutions for four-point boundary value problem of fractional delay differential equations with p-Laplacian operator","volume":"165","author":"Li","year":"2021","journal-title":"Appl. Numer. Math."},{"key":"ref_27","unstructured":"Ulam, S. (1906). A Collection of Mathematical Problems. Interscience Tracts in Pure and Applied Mathmatics, Interscience."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2222","DOI":"10.1073\/pnas.27.4.222","article-title":"On the stability of the linear functional equation","volume":"27","author":"Hyers","year":"1941","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2681","DOI":"10.1007\/s40840-018-0625-x","article-title":"On Ulam\u2019s stability for a coupled systems of nonlinear implicit fractional differential equations","volume":"42","author":"Ali","year":"2019","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Ahmad, M., Zada, A., Ghaderi, M., George, R., and Rezapour, S. (2022). On the existence and stability of a neutral stochastic fractional differential system. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6040203"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"108949","DOI":"10.1016\/j.spl.2020.108949","article-title":"Ulam-Hyers stability of Caputo type fractional stochastic neutral differential equations","volume":"168","author":"Ahmadova","year":"2021","journal-title":"Stat. Probabil. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"112317","DOI":"10.1016\/j.chaos.2022.112317","article-title":"Existence results and Ulam type stability for conformable fractional oscillating system with pure delay","volume":"161","author":"Li","year":"2022","journal-title":"Chaos Soliton. Fract."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Zhao, K. (2022). Stability of a nonlinear Langevin system of ML-Type fractional derivative affected by time-varying delays and differential feedback control. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6120725"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1186\/s13661-023-01785-4","article-title":"Existence theory and Ulam\u2019s stabilities for switched coupled system of implicit impulsive fractional order Langevin equations","volume":"2023","author":"Rizwan","year":"2023","journal-title":"Bound. Value Probl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"173","DOI":"10.3934\/eect.2023042","article-title":"Hyers-Ulam stability result for hilfer fractional integrodifferential stochastic equations with fractional noises and non-instantaneous impulses","volume":"13","author":"Priyadharsini","year":"2024","journal-title":"Evol. Equ. Control. Theory"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Thabet, S., Vivas-Cortez, M., Kedim, I., Samei, M.E., and Ayari, M.I. (2023). Solvability of a \u03f1-Hilfer fractional snap dynamic system on unbounded domains. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7080607"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"17941","DOI":"10.1002\/mma.9539","article-title":"Existence, uniqueness, and stability analysis of fractional Langevin equations with anti-periodic boundary conditions","volume":"46","author":"Shah","year":"2023","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"13351","DOI":"10.3934\/math.2023676","article-title":"Solvability and GUH-stability of a nonlinear CF-fractional coupled Laplacian equations","volume":"8","author":"Zhao","year":"2023","journal-title":"AIMS Math."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"12711","DOI":"10.1002\/mma.9208","article-title":"Analysis of q-fractional coupled implicit systems involving the nonlocal Riemann\u2013Liouville and Erdelyi-Kober q-fractional integral conditions","volume":"46","author":"Alam","year":"2023","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"27840","DOI":"10.3934\/math.20231424","article-title":"Stability analysis of new generalized mean-square stochastic fractional differential equations and their applications in technology","volume":"8","author":"Khan","year":"2023","journal-title":"AIMS Math."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Kiskinov, H., Milev, M., Cholakov, S., and Zahariev, A. (2024). Fundamental matrix, integral representation and stability analysis of the solutions of neutral fractional systems with derivatives in the Riemann\u2013Liouville sense. Fractal Fract., 8.","DOI":"10.3390\/fractalfract8040195"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/s13324-024-00890-6","article-title":"Beam deflection coupled systems of fractional differential equations: Existence of solutions, Ulam-Hyers stability and travelling waves","volume":"14","author":"Bensassa","year":"2024","journal-title":"Anal. Math. Phys."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1186\/s13660-023-03010-3","article-title":"Generalized UH-stability of a nonlinear fractional coupling (p1,p2)-Laplacian system concerned with nonsingular Atangana-Baleanu fractional calculus","volume":"2023","author":"Zhao","year":"2023","journal-title":"J. Inequal. Appl."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1771","DOI":"10.1007\/s12190-024-02033-3","article-title":"Sufficient criteria for the existence of solution to nonlinear fractal-fractional order coupled system with coupled integral boundary conditions","volume":"70","author":"Shah","year":"2024","journal-title":"J. Appl. Math. Comput."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1543","DOI":"10.1007\/s12190-024-02017-3","article-title":"Piecewise conformable fractional impulsive differential system with delay: Existence, uniqueness and Ulam stability","volume":"70","author":"Zhang","year":"2024","journal-title":"J. Appl. Math. Comput."},{"key":"ref_46","unstructured":"Guo, D., and Lakshmikantham, V. (1988). Nonlinear Problems in Abstract Cone, Academic Press."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Zhao, K., Liu, J., and Lv, X. (2024). A unified approach to solvability and stability of multipoint BVPs for Langevin and Sturm\u2013Liouville equations with CH\u2013fractional derivatives and impulses via coincidence theory. Fractal Fract., 8.","DOI":"10.3390\/fractalfract8020111"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"20752","DOI":"10.3934\/math.20221137","article-title":"Global stability of a novel nonlinear diffusion online game addiction model with unsustainable control","volume":"7","author":"Zhao","year":"2022","journal-title":"AIMS Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/6\/774\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:01:38Z","timestamp":1760108498000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/6\/774"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,20]]},"references-count":48,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["sym16060774"],"URL":"https:\/\/doi.org\/10.3390\/sym16060774","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,6,20]]}}}